The aircraft price varies by their engine type, power, fuel capacity, speed, weight, distance range and other parameters. Prices for 518 different aircraft are given in aircraft_price.csv file. The dataset includes 517 aircraft records with 14 quantitative variables and 2 qualitative variables. Quantitative Variables are measured on a continuous ratio scale (e.g., speed in knots, weight in pounds, price in dollars) and Qualitative Variables are measured on a nominal scale (e.g., engine type as piston or other). The dataset contains a variety of aircraft performance and specification data, which could be useful for price prediction.
The explained variable is:
The explanatory variables are:
model_name: Name of the aircraft model.
engine_type: Type of the engine. (Qualitative variable)
engine_power: Power of the engine. (hp)
max_speed: Maximum speed of the aircraft. (Knots(kt))
cruise_speed: Cruise speed of the aircraft. (Knots(kt))
stall_speed: Minimum speed of the aircraft to prevent stalling.(Knots(kt))
fuel_tank: Fuel tank capacity of the aircraft. (gallons(gal))
all_eng_roc: All Engine Rate of Climb. The maximum altitude of the aircraft at full power. (feet/min)
out_eng_roc: Out Engine Rate of Climb. The maximum altitude of the aircraft at out power. (feet/min)
takeoff_distance: The minimum distance required for an aircraft to take off. (feet)
landing_distance: The minimum distance required for an aircraft to land. (feet)
empty_weight: Empty weight of the aircraft. (pounds(lbs))
length: Length of the aircraft. (inch)
wing_span: Wingspan of the aircraft. (inch)
range: Range of the aircraft. (nmi)
# Read the aircraft data from the csv file
Aircraft = read.csv("/Users/anithajoseph/Documents/UofC/DATA603/project/aircraft_price.csv")
print(head(Aircraft))## model_name engine_type engine_power max_speed
## 1 100 Darter (S.L. Industries) Piston 145 104
## 2 7 CCM Champ Piston 85 89
## 3 100 Darter (S.L. Industries) Piston 90 90
## 4 7 AC Champ Piston 85 88
## 5 100 Darter (S.L. Industries) Piston 65 83
## 6 PA-60-700P Aerostar (preliminary) Piston 65 78
## cruise_speed stall_speed fuel_tank all_eng_roc out_eng_roc takeoff_distance
## 1 91 46 36 450 900 1300
## 2 83 44 15 600 720 800
## 3 78 37 19 650 475 850
## 4 78 37 19 620 500 850
## 5 74 33 14 370 632 885
## 6 72 33 15 360 583 880
## landing_distance empty_weight length wing_span range price
## 1 2050 1180 303 449 370 1300000
## 2 1350 820 247 433 190 1230000
## 3 1300 810 257 420 210 1600000
## 4 1300 800 257 420 210 1300000
## 5 1220 740 257 420 175 1250000
## 6 1250 786 244 433 180 1100000Check the Aircraft data for any missing values and then remove rows with missing data to create a cleaned version of the data set. It ensures that the cleaned Aircraft data has complete information in every row so you can analyze it without any gaps in data.
# Check for missing values
sum(is.na(Aircraft))## [1] 10# Remove rows with any missing values
AircraftData <- na.omit(Aircraft)
print(head(AircraftData))## model_name engine_type engine_power max_speed
## 1 100 Darter (S.L. Industries) Piston 145 104
## 2 7 CCM Champ Piston 85 89
## 3 100 Darter (S.L. Industries) Piston 90 90
## 4 7 AC Champ Piston 85 88
## 5 100 Darter (S.L. Industries) Piston 65 83
## 6 PA-60-700P Aerostar (preliminary) Piston 65 78
## cruise_speed stall_speed fuel_tank all_eng_roc out_eng_roc takeoff_distance
## 1 91 46 36 450 900 1300
## 2 83 44 15 600 720 800
## 3 78 37 19 650 475 850
## 4 78 37 19 620 500 850
## 5 74 33 14 370 632 885
## 6 72 33 15 360 583 880
## landing_distance empty_weight length wing_span range price
## 1 2050 1180 303 449 370 1300000
## 2 1350 820 247 433 190 1230000
## 3 1300 810 257 420 210 1600000
## 4 1300 800 257 420 210 1300000
## 5 1220 740 257 420 175 1250000
## 6 1250 786 244 433 180 1100000First order model for predicting the Aircraft price is generated with the all the explanatory variables excluding the model name.
firstordermodel = lm(price ~factor(engine_type)+engine_power +
max_speed + cruise_speed +stall_speed +
fuel_tank + all_eng_roc + out_eng_roc +
takeoff_distance + landing_distance +
empty_weight + length + wing_span + range,
data = AircraftData)
summary(firstordermodel)##
## Call:
## lm(formula = price ~ factor(engine_type) + engine_power + max_speed +
## cruise_speed + stall_speed + fuel_tank + all_eng_roc + out_eng_roc +
## takeoff_distance + landing_distance + empty_weight + length +
## wing_span + range, data = AircraftData)
##
## Residuals:
## Min 1Q Median 3Q Max
## -935282 -228459 -47954 191710 1777937
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) -1.653e+04 2.464e+05 -0.067 0.946532
## factor(engine_type)Piston -4.274e+05 1.255e+05 -3.406 0.000714 ***
## factor(engine_type)Propjet -3.642e+05 1.246e+05 -2.923 0.003629 **
## engine_power 8.070e+01 4.566e+01 1.767 0.077814 .
## max_speed 1.794e+03 3.013e+02 5.954 5e-09 ***
## cruise_speed 4.828e+03 4.782e+02 10.096 < 2e-16 ***
## stall_speed 4.041e+03 2.019e+03 2.001 0.045938 *
## fuel_tank 1.166e+01 1.986e+01 0.587 0.557400
## all_eng_roc 8.712e+00 1.766e+01 0.493 0.622036
## out_eng_roc -6.528e+01 3.569e+01 -1.829 0.067983 .
## takeoff_distance 8.075e+01 5.007e+01 1.613 0.107475
## landing_distance -8.968e+01 1.923e+01 -4.664 4e-06 ***
## empty_weight 6.497e+01 3.208e+01 2.025 0.043403 *
## length 1.533e+03 5.108e+02 3.001 0.002827 **
## wing_span 1.412e+03 4.032e+02 3.503 0.000502 ***
## range 1.825e+02 5.045e+01 3.617 0.000329 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 348000 on 491 degrees of freedom
## Multiple R-squared: 0.8868, Adjusted R-squared: 0.8833
## F-statistic: 256.3 on 15 and 491 DF, p-value: < 2.2e-16Multicollinearity occurs when independent variables are highly correlated, which can distort the coefficients in a regression model. We are testing for Multicollinearity with the following techniques:
Pairs plot
Variance Inflation Factors (VIF).(VIF > 5 → Moderate multicollinearity. VIF > 10 → Severe multicollinearity.)
Let’s plot the most correlated predictors:
pairs(~ price + engine_power + max_speed + cruise_speed +stall_speed +
fuel_tank + all_eng_roc + out_eng_roc + takeoff_distance + landing_distance +
empty_weight + length + wing_span + range, data = AircraftData)
From the above output we can see that there exists correlation for engine_power with landing_distance, empty_weight and fuel_tank respectively. Also there exists a correlation between wing_span and length.
So let’s find the VIF for the variables.
#2. Variance Inflation Factors (VIF)
imcdiag(firstordermodel, method="VIF")##
## Call:
## imcdiag(mod = firstordermodel, method = "VIF")
##
##
## VIF Multicollinearity Diagnostics
##
## VIF detection
## factor(engine_type)Piston 12.5070 1
## factor(engine_type)Propjet 6.5843 0
## engine_power 27.9138 1
## max_speed 4.9745 0
## cruise_speed 10.0485 1
## stall_speed 4.5704 0
## fuel_tank 30.5244 1
## all_eng_roc 2.7720 0
## out_eng_roc 6.7501 0
## takeoff_distance 5.2539 0
## landing_distance 165.2845 1
## empty_weight 138.6177 1
## length 21.4960 1
## wing_span 7.1323 0
## range 5.1477 0
##
## Multicollinearity may be due to factor(engine_type)Piston engine_power cruise_speed fuel_tank landing_distance empty_weight length regressors
##
## 1 --> COLLINEARITY is detected by the test
## 0 --> COLLINEARITY is not detected by the test
##
## ===================================vif(firstordermodel)## GVIF Df GVIF^(1/(2*Df))
## factor(engine_type) 16.141293 2 2.004401
## engine_power 27.913786 1 5.283350
## max_speed 4.974526 1 2.230365
## cruise_speed 10.048520 1 3.169940
## stall_speed 4.570406 1 2.137851
## fuel_tank 30.524423 1 5.524891
## all_eng_roc 2.772011 1 1.664936
## out_eng_roc 6.750085 1 2.598093
## takeoff_distance 5.253891 1 2.292137
## landing_distance 165.284479 1 12.856301
## empty_weight 138.617698 1 11.773602
## length 21.496040 1 4.636382
## wing_span 7.132325 1 2.670641
## range 5.147730 1 2.268861From the output we can see that landing_distance has highest VIF. According to the VIF values, we can list the highest multicollinearity items as below:
Severe Multicollinearity (VIF/GVIF > 10):
landing_distance (VIF=165.28, GVIF=165.28)
empty_weight (VIF=138.62, GVIF=138.62)
fuel_tank (VIF=30.52, GVIF=30.52)
engine_power (VIF=27.91, GVIF=27.91)
length (VIF=21.50, GVIF=21.50)
factor(engine_type)Piston (VIF=12.51, GVIF=16.14)
cruise_speed (VIF=10.05, GVIF=10.05)
Moderate/Low Collinearity (VIF/GVIF < 10): All other variables (e.g.,max_speed, wing_span, range) are safe to retain. GVIF (Generalized Variance Inflation Factor) adjusts for categorical predictors like engine_type. For categorical variables with Df > 1, e.g., engine_type has 2 levels, GVIF^(1/(2*Df)) is interpreted. Here, factor(engine_type) has GVIF^(1/4) = 2.00 → Acceptable (under common thresholds).
So remove the most correlated variables Priority Order:
First: landing_distance (VIF=165.28)
Second: empty_weight (VIF=138.62)
Third: fuel_tank (VIF=30.52)
Fourth: length (VIF=21.50)
Since enigne_power, Cruise_speed and engine_type are statistically and theoretically significant, we cannot remove them. Below code removes the variables in the highest collinearity order.
#firstordermodel
model1 = lm(price ~factor(engine_type)+engine_power + max_speed + cruise_speed +stall_speed +
fuel_tank + all_eng_roc + out_eng_roc + takeoff_distance + landing_distance +
empty_weight + length + wing_span + range, data = AircraftData)
summary(model1)##
## Call:
## lm(formula = price ~ factor(engine_type) + engine_power + max_speed +
## cruise_speed + stall_speed + fuel_tank + all_eng_roc + out_eng_roc +
## takeoff_distance + landing_distance + empty_weight + length +
## wing_span + range, data = AircraftData)
##
## Residuals:
## Min 1Q Median 3Q Max
## -935282 -228459 -47954 191710 1777937
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) -1.653e+04 2.464e+05 -0.067 0.946532
## factor(engine_type)Piston -4.274e+05 1.255e+05 -3.406 0.000714 ***
## factor(engine_type)Propjet -3.642e+05 1.246e+05 -2.923 0.003629 **
## engine_power 8.070e+01 4.566e+01 1.767 0.077814 .
## max_speed 1.794e+03 3.013e+02 5.954 5e-09 ***
## cruise_speed 4.828e+03 4.782e+02 10.096 < 2e-16 ***
## stall_speed 4.041e+03 2.019e+03 2.001 0.045938 *
## fuel_tank 1.166e+01 1.986e+01 0.587 0.557400
## all_eng_roc 8.712e+00 1.766e+01 0.493 0.622036
## out_eng_roc -6.528e+01 3.569e+01 -1.829 0.067983 .
## takeoff_distance 8.075e+01 5.007e+01 1.613 0.107475
## landing_distance -8.968e+01 1.923e+01 -4.664 4e-06 ***
## empty_weight 6.497e+01 3.208e+01 2.025 0.043403 *
## length 1.533e+03 5.108e+02 3.001 0.002827 **
## wing_span 1.412e+03 4.032e+02 3.503 0.000502 ***
## range 1.825e+02 5.045e+01 3.617 0.000329 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 348000 on 491 degrees of freedom
## Multiple R-squared: 0.8868, Adjusted R-squared: 0.8833
## F-statistic: 256.3 on 15 and 491 DF, p-value: < 2.2e-16imcdiag(model1, method="VIF")##
## Call:
## imcdiag(mod = model1, method = "VIF")
##
##
## VIF Multicollinearity Diagnostics
##
## VIF detection
## factor(engine_type)Piston 12.5070 1
## factor(engine_type)Propjet 6.5843 0
## engine_power 27.9138 1
## max_speed 4.9745 0
## cruise_speed 10.0485 1
## stall_speed 4.5704 0
## fuel_tank 30.5244 1
## all_eng_roc 2.7720 0
## out_eng_roc 6.7501 0
## takeoff_distance 5.2539 0
## landing_distance 165.2845 1
## empty_weight 138.6177 1
## length 21.4960 1
## wing_span 7.1323 0
## range 5.1477 0
##
## Multicollinearity may be due to factor(engine_type)Piston engine_power cruise_speed fuel_tank landing_distance empty_weight length regressors
##
## 1 --> COLLINEARITY is detected by the test
## 0 --> COLLINEARITY is not detected by the test
##
## ===================================#removed landing_distance
model2 = lm(price ~factor(engine_type)+engine_power + max_speed + cruise_speed +stall_speed +
fuel_tank + all_eng_roc + out_eng_roc + takeoff_distance +
empty_weight + length + wing_span + range, data = AircraftData)
summary(model2)##
## Call:
## lm(formula = price ~ factor(engine_type) + engine_power + max_speed +
## cruise_speed + stall_speed + fuel_tank + all_eng_roc + out_eng_roc +
## takeoff_distance + empty_weight + length + wing_span + range,
## data = AircraftData)
##
## Residuals:
## Min 1Q Median 3Q Max
## -922389 -230292 -50931 200933 1993660
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 57568.57 251044.88 0.229 0.818719
## factor(engine_type)Piston -448395.55 128035.27 -3.502 0.000504 ***
## factor(engine_type)Propjet -394540.95 127027.84 -3.106 0.002006 **
## engine_power 46.97 46.03 1.020 0.308014
## max_speed 1640.00 305.71 5.365 1.25e-07 ***
## cruise_speed 4772.82 488.07 9.779 < 2e-16 ***
## stall_speed 4845.70 2053.93 2.359 0.018703 *
## fuel_tank -19.76 19.07 -1.036 0.300587
## all_eng_roc 8.86 18.03 0.491 0.623346
## out_eng_roc -54.06 36.35 -1.487 0.137661
## takeoff_distance 57.17 50.86 1.124 0.261460
## empty_weight -60.34 17.90 -3.370 0.000809 ***
## length 1420.64 520.91 2.727 0.006615 **
## wing_span 1302.44 410.90 3.170 0.001621 **
## range 185.46 51.50 3.601 0.000349 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 355300 on 492 degrees of freedom
## Multiple R-squared: 0.8817, Adjusted R-squared: 0.8784
## F-statistic: 262 on 14 and 492 DF, p-value: < 2.2e-16imcdiag(model2, method="VIF")##
## Call:
## imcdiag(mod = model2, method = "VIF")
##
##
## VIF Multicollinearity Diagnostics
##
## VIF detection
## factor(engine_type)Piston 12.4909 1
## factor(engine_type)Propjet 6.5664 0
## engine_power 27.2139 1
## max_speed 4.9151 0
## cruise_speed 10.0423 1
## stall_speed 4.5370 0
## fuel_tank 27.0119 1
## all_eng_roc 2.7720 0
## out_eng_roc 6.7194 0
## takeoff_distance 5.2004 0
## empty_weight 41.4192 1
## length 21.4482 1
## wing_span 7.1080 0
## range 5.1469 0
##
## Multicollinearity may be due to factor(engine_type)Piston engine_power cruise_speed fuel_tank empty_weight length regressors
##
## 1 --> COLLINEARITY is detected by the test
## 0 --> COLLINEARITY is not detected by the test
##
## ===================================#removed empty_weight
model3 = lm(price ~factor(engine_type)+engine_power + max_speed + cruise_speed +stall_speed +
fuel_tank + all_eng_roc + out_eng_roc + takeoff_distance + length + wing_span + range,
data = AircraftData)
summary(model3)##
## Call:
## lm(formula = price ~ factor(engine_type) + engine_power + max_speed +
## cruise_speed + stall_speed + fuel_tank + all_eng_roc + out_eng_roc +
## takeoff_distance + length + wing_span + range, data = AircraftData)
##
## Residuals:
## Min 1Q Median 3Q Max
## -1010722 -228791 -53279 207544 1884293
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 569816.12 201901.75 2.822 0.00496 **
## factor(engine_type)Piston -613204.07 119567.06 -5.129 4.20e-07 ***
## factor(engine_type)Propjet -588560.93 114419.57 -5.144 3.89e-07 ***
## engine_power -27.66 40.78 -0.678 0.49794
## max_speed 1494.70 305.82 4.888 1.38e-06 ***
## cruise_speed 4620.14 491.04 9.409 < 2e-16 ***
## stall_speed 2996.44 1999.99 1.498 0.13471
## fuel_tank -50.73 16.89 -3.004 0.00280 **
## all_eng_roc 14.36 18.14 0.791 0.42916
## out_eng_roc -37.31 36.39 -1.025 0.30571
## takeoff_distance 41.29 51.17 0.807 0.42006
## length 921.06 504.59 1.825 0.06855 .
## wing_span 894.18 396.75 2.254 0.02465 *
## range 207.97 51.60 4.031 6.44e-05 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 359000 on 493 degrees of freedom
## Multiple R-squared: 0.879, Adjusted R-squared: 0.8758
## F-statistic: 275.5 on 13 and 493 DF, p-value: < 2.2e-16imcdiag(model3, method="VIF")##
## Call:
## imcdiag(mod = model3, method = "VIF")
##
##
## VIF Multicollinearity Diagnostics
##
## VIF detection
## factor(engine_type)Piston 10.6691 1
## factor(engine_type)Propjet 5.2179 0
## engine_power 20.9170 1
## max_speed 4.8173 0
## cruise_speed 9.9558 0
## stall_speed 4.2133 0
## fuel_tank 20.7414 1
## all_eng_roc 2.7493 0
## out_eng_roc 6.5939 0
## takeoff_distance 5.1557 0
## length 19.7116 1
## wing_span 6.4903 0
## range 5.0603 0
##
## Multicollinearity may be due to factor(engine_type)Piston engine_power fuel_tank length regressors
##
## 1 --> COLLINEARITY is detected by the test
## 0 --> COLLINEARITY is not detected by the test
##
## ===================================#removed fuel_tank
model4 = lm(price ~factor(engine_type)+engine_power + max_speed + cruise_speed +stall_speed +
all_eng_roc + out_eng_roc + takeoff_distance + length + wing_span + range,
data = AircraftData)
summary(model4)##
## Call:
## lm(formula = price ~ factor(engine_type) + engine_power + max_speed +
## cruise_speed + stall_speed + all_eng_roc + out_eng_roc +
## takeoff_distance + length + wing_span + range, data = AircraftData)
##
## Residuals:
## Min 1Q Median 3Q Max
## -1310103 -225685 -48704 196659 1939134
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 695908.97 199089.37 3.495 0.000516 ***
## factor(engine_type)Piston -641222.25 120167.30 -5.336 1.45e-07 ***
## factor(engine_type)Propjet -560763.09 114967.61 -4.878 1.45e-06 ***
## engine_power -126.51 24.28 -5.211 2.76e-07 ***
## max_speed 1467.85 308.16 4.763 2.51e-06 ***
## cruise_speed 4637.80 494.98 9.370 < 2e-16 ***
## stall_speed 3901.87 1993.15 1.958 0.050834 .
## all_eng_roc 18.96 18.22 1.040 0.298814
## out_eng_roc -60.79 35.83 -1.697 0.090357 .
## takeoff_distance 73.95 50.40 1.467 0.142949
## length 954.72 508.55 1.877 0.061062 .
## wing_span 622.99 389.47 1.600 0.110330
## range 154.44 48.82 3.164 0.001654 **
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 361900 on 494 degrees of freedom
## Multiple R-squared: 0.8768, Adjusted R-squared: 0.8738
## F-statistic: 293 on 12 and 494 DF, p-value: < 2.2e-16imcdiag(model4, method="VIF")##
## Call:
## imcdiag(mod = model4, method = "VIF")
##
##
## VIF Multicollinearity Diagnostics
##
## VIF detection
## factor(engine_type)Piston 10.6042 1
## factor(engine_type)Propjet 5.1838 0
## engine_power 7.2957 0
## max_speed 4.8132 0
## cruise_speed 9.9544 0
## stall_speed 4.1176 0
## all_eng_roc 2.7298 0
## out_eng_roc 6.2897 0
## takeoff_distance 4.9230 0
## length 19.7019 1
## wing_span 6.1544 0
## range 4.4570 0
##
## Multicollinearity may be due to factor(engine_type)Piston length regressors
##
## 1 --> COLLINEARITY is detected by the test
## 0 --> COLLINEARITY is not detected by the test
##
## ===================================#removed length
model5 = lm(price ~factor(engine_type)+engine_power + max_speed + cruise_speed +stall_speed +
all_eng_roc + out_eng_roc + takeoff_distance + wing_span + range, data = AircraftData)
summary(model5)##
## Call:
## lm(formula = price ~ factor(engine_type) + engine_power + max_speed +
## cruise_speed + stall_speed + all_eng_roc + out_eng_roc +
## takeoff_distance + wing_span + range, data = AircraftData)
##
## Residuals:
## Min 1Q Median 3Q Max
## -1291854 -224274 -51822 188629 2036117
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 723802.03 199039.77 3.636 0.000305 ***
## factor(engine_type)Piston -670177.36 119476.86 -5.609 3.38e-08 ***
## factor(engine_type)Propjet -547492.38 115042.31 -4.759 2.56e-06 ***
## engine_power -107.82 22.20 -4.858 1.60e-06 ***
## max_speed 1522.64 307.56 4.951 1.02e-06 ***
## cruise_speed 4702.09 495.05 9.498 < 2e-16 ***
## stall_speed 4695.48 1952.76 2.405 0.016559 *
## all_eng_roc 21.87 18.20 1.202 0.230105
## out_eng_roc -44.91 34.90 -1.287 0.198788
## takeoff_distance 75.07 50.53 1.486 0.137965
## wing_span 1094.87 298.26 3.671 0.000268 ***
## range 166.44 48.52 3.430 0.000653 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 362800 on 495 degrees of freedom
## Multiple R-squared: 0.8759, Adjusted R-squared: 0.8732
## F-statistic: 317.7 on 11 and 495 DF, p-value: < 2.2e-16imcdiag(model5, method="VIF")##
## Call:
## imcdiag(mod = model5, method = "VIF")
##
##
## VIF Multicollinearity Diagnostics
##
## VIF detection
## factor(engine_type)Piston 10.4295 1
## factor(engine_type)Propjet 5.1642 0
## engine_power 6.0686 0
## max_speed 4.7700 0
## cruise_speed 9.9067 0
## stall_speed 3.9324 0
## all_eng_roc 2.7099 0
## out_eng_roc 5.9389 0
## takeoff_distance 4.9223 0
## wing_span 3.5910 0
## range 4.3806 0
##
## Multicollinearity may be due to factor(engine_type)Piston regressors
##
## 1 --> COLLINEARITY is detected by the test
## 0 --> COLLINEARITY is not detected by the test
##
## ===================================# removed insignifocant terms:all_eng_roc, out_eng_roc & takeoff_distance from model5
model6 = lm(price ~factor(engine_type)+engine_power + max_speed + cruise_speed +stall_speed +
wing_span + range, data = AircraftData)
summary(model6)##
## Call:
## lm(formula = price ~ factor(engine_type) + engine_power + max_speed +
## cruise_speed + stall_speed + wing_span + range, data = AircraftData)
##
## Residuals:
## Min 1Q Median 3Q Max
## -1305416 -227626 -54056 193438 2121608
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 693659.32 186978.43 3.710 0.000231 ***
## factor(engine_type)Piston -661350.47 111533.27 -5.930 5.68e-09 ***
## factor(engine_type)Propjet -527735.55 110064.68 -4.795 2.15e-06 ***
## engine_power -115.72 21.27 -5.441 8.31e-08 ***
## max_speed 1571.61 305.17 5.150 3.76e-07 ***
## cruise_speed 4963.33 474.50 10.460 < 2e-16 ***
## stall_speed 5401.47 1817.88 2.971 0.003109 **
## wing_span 1104.12 288.70 3.825 0.000148 ***
## range 159.60 47.52 3.359 0.000843 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 363200 on 498 degrees of freedom
## Multiple R-squared: 0.8749, Adjusted R-squared: 0.8729
## F-statistic: 435.3 on 8 and 498 DF, p-value: < 2.2e-16imcdiag(model6, method="VIF")##
## Call:
## imcdiag(mod = model6, method = "VIF")
##
##
## VIF Multicollinearity Diagnostics
##
## VIF detection
## factor(engine_type)Piston 9.0677 0
## factor(engine_type)Propjet 4.7161 0
## engine_power 5.5579 0
## max_speed 4.6856 0
## cruise_speed 9.0803 0
## stall_speed 3.4000 0
## wing_span 3.3566 0
## range 4.1923 0
##
## NOTE: VIF Method Failed to detect multicollinearity
##
##
## 0 --> COLLINEARITY is not detected by the test
##
## ===================================# removed cruise_speed: even if the multicollinearity =0 , VIF>5, so removed from model5
model7 = lm(price ~factor(engine_type)+engine_power + max_speed + stall_speed + all_eng_roc + out_eng_roc +
takeoff_distance + wing_span + range, data = AircraftData)
summary(model7)##
## Call:
## lm(formula = price ~ factor(engine_type) + engine_power + max_speed +
## stall_speed + all_eng_roc + out_eng_roc + takeoff_distance +
## wing_span + range, data = AircraftData)
##
## Residuals:
## Min 1Q Median 3Q Max
## -1352143 -252725 -53828 229130 1554800
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 1.390e+06 2.023e+05 6.868 1.96e-11 ***
## factor(engine_type)Piston -1.207e+06 1.144e+05 -10.551 < 2e-16 ***
## factor(engine_type)Propjet -1.019e+06 1.127e+05 -9.035 < 2e-16 ***
## engine_power -1.130e+02 2.410e+01 -4.686 3.60e-06 ***
## max_speed 2.389e+03 3.191e+02 7.487 3.24e-13 ***
## stall_speed 1.040e+04 2.018e+03 5.151 3.74e-07 ***
## all_eng_roc 6.346e+01 1.919e+01 3.306 0.00101 **
## out_eng_roc -6.729e+01 3.782e+01 -1.779 0.07587 .
## takeoff_distance 1.503e+02 5.420e+01 2.774 0.00575 **
## wing_span 9.717e+02 3.237e+02 3.002 0.00282 **
## range 2.791e+02 5.111e+01 5.460 7.53e-08 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 394100 on 496 degrees of freedom
## Multiple R-squared: 0.8533, Adjusted R-squared: 0.8503
## F-statistic: 288.5 on 10 and 496 DF, p-value: < 2.2e-16#imcdiag(model7, method="VIF")
# removed insignifocant terms:out_eng_roc from model6
model8 = lm(price ~factor(engine_type)+engine_power + max_speed + stall_speed +
all_eng_roc + takeoff_distance + wing_span + range, data = AircraftData)
summary(model8)##
## Call:
## lm(formula = price ~ factor(engine_type) + engine_power + max_speed +
## stall_speed + all_eng_roc + takeoff_distance + wing_span +
## range, data = AircraftData)
##
## Residuals:
## Min 1Q Median 3Q Max
## -1380009 -248912 -51757 236662 1520550
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 1.282e+06 1.934e+05 6.625 9.02e-11 ***
## factor(engine_type)Piston -1.144e+06 1.091e+05 -10.489 < 2e-16 ***
## factor(engine_type)Propjet -9.659e+05 1.090e+05 -8.860 < 2e-16 ***
## engine_power -1.240e+02 2.333e+01 -5.316 1.61e-07 ***
## max_speed 2.383e+03 3.197e+02 7.453 4.08e-13 ***
## stall_speed 1.020e+04 2.020e+03 5.051 6.19e-07 ***
## all_eng_roc 6.615e+01 1.918e+01 3.449 0.00061 ***
## takeoff_distance 1.001e+02 4.636e+01 2.159 0.03134 *
## wing_span 1.036e+03 3.224e+02 3.213 0.00140 **
## range 2.709e+02 5.101e+01 5.310 1.65e-07 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 395000 on 497 degrees of freedom
## Multiple R-squared: 0.8524, Adjusted R-squared: 0.8497
## F-statistic: 318.8 on 9 and 497 DF, p-value: < 2.2e-16imcdiag(model7, method="VIF")##
## Call:
## imcdiag(mod = model7, method = "VIF")
##
##
## VIF Multicollinearity Diagnostics
##
## VIF detection
## factor(engine_type)Piston 8.0988 0
## factor(engine_type)Propjet 4.2041 0
## engine_power 6.0650 0
## max_speed 4.3507 0
## stall_speed 3.5607 0
## all_eng_roc 2.5532 0
## out_eng_roc 5.9119 0
## takeoff_distance 4.8012 0
## wing_span 3.5842 0
## range 4.1191 0
##
## NOTE: VIF Method Failed to detect multicollinearity
##
##
## 0 --> COLLINEARITY is not detected by the test
##
## ===================================Proceeding with model 5, since the multicollinearity removed.
model5 = lm(price ~factor(engine_type)+engine_power + max_speed + cruise_speed +stall_speed +
all_eng_roc + out_eng_roc + takeoff_distance + wing_span + range, data = AircraftData)
summary(model5)##
## Call:
## lm(formula = price ~ factor(engine_type) + engine_power + max_speed +
## cruise_speed + stall_speed + all_eng_roc + out_eng_roc +
## takeoff_distance + wing_span + range, data = AircraftData)
##
## Residuals:
## Min 1Q Median 3Q Max
## -1291854 -224274 -51822 188629 2036117
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 723802.03 199039.77 3.636 0.000305 ***
## factor(engine_type)Piston -670177.36 119476.86 -5.609 3.38e-08 ***
## factor(engine_type)Propjet -547492.38 115042.31 -4.759 2.56e-06 ***
## engine_power -107.82 22.20 -4.858 1.60e-06 ***
## max_speed 1522.64 307.56 4.951 1.02e-06 ***
## cruise_speed 4702.09 495.05 9.498 < 2e-16 ***
## stall_speed 4695.48 1952.76 2.405 0.016559 *
## all_eng_roc 21.87 18.20 1.202 0.230105
## out_eng_roc -44.91 34.90 -1.287 0.198788
## takeoff_distance 75.07 50.53 1.486 0.137965
## wing_span 1094.87 298.26 3.671 0.000268 ***
## range 166.44 48.52 3.430 0.000653 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 362800 on 495 degrees of freedom
## Multiple R-squared: 0.8759, Adjusted R-squared: 0.8732
## F-statistic: 317.7 on 11 and 495 DF, p-value: < 2.2e-16imcdiag(model5, method="VIF")##
## Call:
## imcdiag(mod = model5, method = "VIF")
##
##
## VIF Multicollinearity Diagnostics
##
## VIF detection
## factor(engine_type)Piston 10.4295 1
## factor(engine_type)Propjet 5.1642 0
## engine_power 6.0686 0
## max_speed 4.7700 0
## cruise_speed 9.9067 0
## stall_speed 3.9324 0
## all_eng_roc 2.7099 0
## out_eng_roc 5.9389 0
## takeoff_distance 4.9223 0
## wing_span 3.5910 0
## range 4.3806 0
##
## Multicollinearity may be due to factor(engine_type)Piston regressors
##
## 1 --> COLLINEARITY is detected by the test
## 0 --> COLLINEARITY is not detected by the test
##
## ===================================plot(~price +factor(engine_type)+engine_power + max_speed + cruise_speed +stall_speed +
all_eng_roc + out_eng_roc + takeoff_distance + wing_span + range, data = AircraftData)
fullmodel = lm(price ~ engine_type + engine_power + max_speed + cruise_speed +stall_speed +
all_eng_roc + out_eng_roc + takeoff_distance + wing_span + range, data = AircraftData)
stepmod=ols_step_both_p(fullmodel,p_enter = 0.05, p_remove = 0.1, details=TRUE)## Stepwise Selection Method
## -------------------------
##
## Candidate Terms:
##
## 1. engine_type
## 2. engine_power
## 3. max_speed
## 4. cruise_speed
## 5. stall_speed
## 6. all_eng_roc
## 7. out_eng_roc
## 8. takeoff_distance
## 9. wing_span
## 10. range
##
##
## Step => 0
## Model => price ~ 1
## R2 => 0
##
## Initiating stepwise selection...
##
## Step => 1
## Selected => cruise_speed
## Model => price ~ cruise_speed
## R2 => 0.84
##
## Step => 2
## Selected => max_speed
## Model => price ~ cruise_speed + max_speed
## R2 => 0.853
##
## Step => 3
## Selected => wing_span
## Model => price ~ cruise_speed + max_speed + wing_span
## R2 => 0.859
##
## Step => 4
## Selected => engine_power
## Model => price ~ cruise_speed + max_speed + wing_span + engine_power
## R2 => 0.863
##
## Step => 5
## Selected => engine_type
## Model => price ~ cruise_speed + max_speed + wing_span + engine_power + engine_type
## R2 => 0.87
##
## Step => 6
## Selected => range
## Model => price ~ cruise_speed + max_speed + wing_span + engine_power + engine_type + range
## R2 => 0.873
##
## Step => 7
## Selected => stall_speed
## Model => price ~ cruise_speed + max_speed + wing_span + engine_power + engine_type + range + stall_speed
## R2 => 0.875
##
##
## No more variables to be added or removed.summary(stepmod$model)##
## Call:
## lm(formula = paste(response, "~", paste(preds, collapse = " + ")),
## data = l)
##
## Residuals:
## Min 1Q Median 3Q Max
## -1305416 -227626 -54056 193438 2121608
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 693659.32 186978.43 3.710 0.000231 ***
## cruise_speed 4963.33 474.50 10.460 < 2e-16 ***
## max_speed 1571.61 305.17 5.150 3.76e-07 ***
## wing_span 1104.12 288.70 3.825 0.000148 ***
## engine_power -115.72 21.27 -5.441 8.31e-08 ***
## engine_typePiston -661350.47 111533.27 -5.930 5.68e-09 ***
## engine_typePropjet -527735.55 110064.68 -4.795 2.15e-06 ***
## range 159.60 47.52 3.359 0.000843 ***
## stall_speed 5401.47 1817.88 2.971 0.003109 **
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 363200 on 498 degrees of freedom
## Multiple R-squared: 0.8749, Adjusted R-squared: 0.8729
## F-statistic: 435.3 on 8 and 498 DF, p-value: < 2.2e-16stepmod$metrics$adj_r2## [1] 0.8400379 0.8519476 0.8576611 0.8620161 0.8684412 0.8708709 0.8728655Best additive model using the Stepwise selection procedure is as follows:
\(\widehat{\text{price}}\) = 693659.32 - 661350.47Xengine_typePiston - 527735.55Xengine_typePropjet - 115.72Xengine_power + 1571.61Xmax_speed + 4963.33Xcruise_speed + 5401.47Xstall_speed + 1104.12Xwing_span + 159.60Xrange
ExecPriceFor=ols_step_forward_p(fullmodel,p_val = 0.05, details=TRUE)## Forward Selection Method
## ------------------------
##
## Candidate Terms:
##
## 1. engine_type
## 2. engine_power
## 3. max_speed
## 4. cruise_speed
## 5. stall_speed
## 6. all_eng_roc
## 7. out_eng_roc
## 8. takeoff_distance
## 9. wing_span
## 10. range
##
##
## Step => 0
## Model => price ~ 1
## R2 => 0
##
## Initiating stepwise selection...
##
## Selection Metrics Table
## ------------------------------------------------------------------------
## Predictor Pr(>|t|) R-Squared Adj. R-Squared AIC
## ------------------------------------------------------------------------
## cruise_speed 0.00000 0.840 0.840 14541.306
## engine_type 0.00000 0.734 0.733 14801.855
## max_speed 0.00000 0.728 0.727 14811.548
## stall_speed 0.00000 0.621 0.620 14979.453
## takeoff_distance 0.00000 0.611 0.610 14993.017
## out_eng_roc 0.00000 0.595 0.594 15013.720
## range 0.00000 0.535 0.534 15082.938
## all_eng_roc 0.00000 0.513 0.512 15106.338
## engine_power 0.00000 0.454 0.453 15164.527
## wing_span 0.00000 0.374 0.372 15234.377
## ------------------------------------------------------------------------
##
## Step => 1
## Selected => cruise_speed
## Model => price ~ cruise_speed
## R2 => 0.84
##
## Selection Metrics Table
## ------------------------------------------------------------------------
## Predictor Pr(>|t|) R-Squared Adj. R-Squared AIC
## ------------------------------------------------------------------------
## max_speed 0.00000 0.853 0.852 14503.075
## engine_type 0.00000 0.853 0.852 14503.419
## stall_speed 0.00000 0.849 0.848 14515.385
## wing_span 0.00000 0.848 0.848 14516.785
## takeoff_distance 0.00000 0.848 0.847 14518.994
## range 2e-05 0.846 0.845 14525.373
## out_eng_roc 0.00747 0.843 0.842 14536.101
## all_eng_roc 0.08229 0.841 0.841 14540.266
## engine_power 0.86979 0.840 0.840 14543.279
## ------------------------------------------------------------------------
##
## Step => 2
## Selected => max_speed
## Model => price ~ cruise_speed + max_speed
## R2 => 0.853
##
## Selection Metrics Table
## ------------------------------------------------------------------------
## Predictor Pr(>|t|) R-Squared Adj. R-Squared AIC
## ------------------------------------------------------------------------
## engine_type 0.00000 0.862 0.861 14474.722
## wing_span 1e-05 0.859 0.858 14484.115
## stall_speed 2e-05 0.858 0.857 14486.964
## takeoff_distance 3e-05 0.857 0.857 14487.716
## range 0.00126 0.856 0.855 14494.583
## out_eng_roc 0.13339 0.853 0.852 14502.802
## engine_power 0.26912 0.853 0.852 14503.842
## all_eng_roc 0.36159 0.853 0.852 14504.235
## ------------------------------------------------------------------------
##
## Step => 3
## Selected => wing_span
## Model => price ~ cruise_speed + max_speed + wing_span
## R2 => 0.859
##
## Selection Metrics Table
## ------------------------------------------------------------------------
## Predictor Pr(>|t|) R-Squared Adj. R-Squared AIC
## ------------------------------------------------------------------------
## engine_power 5e-05 0.863 0.862 14469.351
## engine_type 5e-05 0.864 0.863 14468.015
## stall_speed 0.00097 0.862 0.860 14475.119
## takeoff_distance 0.00211 0.861 0.860 14476.560
## all_eng_roc 0.05289 0.860 0.858 14482.326
## range 0.41669 0.859 0.858 14485.448
## out_eng_roc 0.69134 0.859 0.857 14485.955
## ------------------------------------------------------------------------
##
## Step => 4
## Selected => engine_power
## Model => price ~ cruise_speed + max_speed + wing_span + engine_power
## R2 => 0.863
##
## Selection Metrics Table
## ------------------------------------------------------------------------
## Predictor Pr(>|t|) R-Squared Adj. R-Squared AIC
## ------------------------------------------------------------------------
## engine_type 0.00000 0.870 0.868 14447.152
## takeoff_distance 0.00970 0.865 0.864 14464.574
## range 0.01788 0.865 0.863 14465.670
## stall_speed 0.02123 0.865 0.863 14465.975
## all_eng_roc 0.04258 0.864 0.863 14467.186
## out_eng_roc 0.06035 0.864 0.863 14467.777
## ------------------------------------------------------------------------
##
## Step => 5
## Selected => engine_type
## Model => price ~ cruise_speed + max_speed + wing_span + engine_power + engine_type
## R2 => 0.87
##
## Selection Metrics Table
## ------------------------------------------------------------------------
## Predictor Pr(>|t|) R-Squared Adj. R-Squared AIC
## ------------------------------------------------------------------------
## range 0.00134 0.873 0.871 14438.686
## stall_speed 0.00498 0.872 0.870 14441.132
## takeoff_distance 0.02453 0.871 0.870 14444.009
## all_eng_roc 0.24900 0.870 0.869 14447.800
## out_eng_roc 0.56433 0.870 0.868 14448.814
## ------------------------------------------------------------------------
##
## Step => 6
## Selected => range
## Model => price ~ cruise_speed + max_speed + wing_span + engine_power + engine_type + range
## R2 => 0.873
##
## Selection Metrics Table
## ------------------------------------------------------------------------
## Predictor Pr(>|t|) R-Squared Adj. R-Squared AIC
## ------------------------------------------------------------------------
## stall_speed 0.00311 0.875 0.873 14431.776
## takeoff_distance 0.05944 0.874 0.872 14437.065
## all_eng_roc 0.12034 0.873 0.871 14438.227
## out_eng_roc 0.93557 0.873 0.871 14440.679
## ------------------------------------------------------------------------
##
## Step => 7
## Selected => stall_speed
## Model => price ~ cruise_speed + max_speed + wing_span + engine_power + engine_type + range + stall_speed
## R2 => 0.875
##
## Selection Metrics Table
## ------------------------------------------------------------------------
## Predictor Pr(>|t|) R-Squared Adj. R-Squared AIC
## ------------------------------------------------------------------------
## all_eng_roc 0.20827 0.875 0.873 14432.159
## takeoff_distance 0.35603 0.875 0.873 14432.906
## out_eng_roc 0.50131 0.875 0.873 14433.315
## ------------------------------------------------------------------------
##
##
## No more variables to be added.
##
## Variables Selected:
##
## => cruise_speed
## => max_speed
## => wing_span
## => engine_power
## => engine_type
## => range
## => stall_speedsummary(ExecPriceFor$model)##
## Call:
## lm(formula = paste(response, "~", paste(preds, collapse = " + ")),
## data = l)
##
## Residuals:
## Min 1Q Median 3Q Max
## -1305416 -227626 -54056 193438 2121608
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 693659.32 186978.43 3.710 0.000231 ***
## cruise_speed 4963.33 474.50 10.460 < 2e-16 ***
## max_speed 1571.61 305.17 5.150 3.76e-07 ***
## wing_span 1104.12 288.70 3.825 0.000148 ***
## engine_power -115.72 21.27 -5.441 8.31e-08 ***
## engine_typePiston -661350.47 111533.27 -5.930 5.68e-09 ***
## engine_typePropjet -527735.55 110064.68 -4.795 2.15e-06 ***
## range 159.60 47.52 3.359 0.000843 ***
## stall_speed 5401.47 1817.88 2.971 0.003109 **
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 363200 on 498 degrees of freedom
## Multiple R-squared: 0.8749, Adjusted R-squared: 0.8729
## F-statistic: 435.3 on 8 and 498 DF, p-value: < 2.2e-16ExecPriceFor$metrics$adj_r2## [1] 0.8400379 0.8519476 0.8576611 0.8620161 0.8684412 0.8708709 0.8728655Best additive model using the Forward selection procedure is as follows:
\(\widehat{\text{price}}\) = 693659.32 - 661350.47Xengine_typePiston - 527735.55Xengine_typePropjet - 115.72Xengine_power + 1571.61Xmax_speed + 4963.33Xcruise_speed + 5401.47Xstall_speed + 1104.12Xwing_span + 159.60Xrange
ExecPriceBack=ols_step_backward_p(fullmodel, p_val = 0.05, details=TRUE)## Backward Elimination Method
## ---------------------------
##
## Candidate Terms:
##
## 1. engine_type
## 2. engine_power
## 3. max_speed
## 4. cruise_speed
## 5. stall_speed
## 6. all_eng_roc
## 7. out_eng_roc
## 8. takeoff_distance
## 9. wing_span
## 10. range
##
##
## Step => 0
## Model => price ~ engine_type + engine_power + max_speed + cruise_speed + stall_speed + all_eng_roc + out_eng_roc + takeoff_distance + wing_span + range
## R2 => 0.876
##
## Initiating stepwise selection...
##
## Step => 1
## Removed => all_eng_roc
## Model => price ~ engine_type + engine_power + max_speed + cruise_speed + stall_speed + out_eng_roc + takeoff_distance + wing_span + range
## R2 => 0.87555
##
## Step => 2
## Removed => out_eng_roc
## Model => price ~ engine_type + engine_power + max_speed + cruise_speed + stall_speed + takeoff_distance + wing_span + range
## R2 => 0.87509
##
## Step => 3
## Removed => takeoff_distance
## Model => price ~ engine_type + engine_power + max_speed + cruise_speed + stall_speed + wing_span + range
## R2 => 0.87488
##
##
## No more variables to be removed.
##
## Variables Removed:
##
## => all_eng_roc
## => out_eng_roc
## => takeoff_distancesummary(ExecPriceBack$model)##
## Call:
## lm(formula = paste(response, "~", paste(c(include, cterms), collapse = " + ")),
## data = l)
##
## Residuals:
## Min 1Q Median 3Q Max
## -1305416 -227626 -54056 193438 2121608
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 693659.32 186978.43 3.710 0.000231 ***
## engine_typePiston -661350.47 111533.27 -5.930 5.68e-09 ***
## engine_typePropjet -527735.55 110064.68 -4.795 2.15e-06 ***
## engine_power -115.72 21.27 -5.441 8.31e-08 ***
## max_speed 1571.61 305.17 5.150 3.76e-07 ***
## cruise_speed 4963.33 474.50 10.460 < 2e-16 ***
## stall_speed 5401.47 1817.88 2.971 0.003109 **
## wing_span 1104.12 288.70 3.825 0.000148 ***
## range 159.60 47.52 3.359 0.000843 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 363200 on 498 degrees of freedom
## Multiple R-squared: 0.8749, Adjusted R-squared: 0.8729
## F-statistic: 435.3 on 8 and 498 DF, p-value: < 2.2e-16Best additive model using the Backwards elimination procedure is as follows:
\(\widehat{\text{price}}\) = 693659.32 - 661350.47Xengine_typePiston - 527735.55Xengine_typePropjet - 115.72Xengine_power + 1571.61Xmax_speed + 4963.33Xcruise_speed + 5401.47Xstall_speed + 1104.12Xwing_span + 159.60Xrange
ExecPriceBack$metrics## step variable r2 adj_r2 aic sbc sbic
## 1 1 all_eng_roc 0.8755545 0.8730456 14433.02 14483.76 12992.64
## 2 2 out_eng_roc 0.8750900 0.8728280 14432.91 14479.42 12992.42
## 3 3 takeoff_distance 0.8748755 0.8728655 14431.78 14474.06 12991.22
## mallows_cp rmse
## 1 11.44377 359021.5
## 2 11.29699 359691.0
## 3 10.15265 359999.7ExecPriceBack$metrics$adj_r2## [1] 0.8730456 0.8728280 0.8728655#option 1
ExecSubsets=ols_step_best_subset(fullmodel, details=TRUE)
ExecSubsets$metrics## mindex n
## 1 1 1
## 2 2 2
## 3 3 3
## 4 4 4
## 5 5 5
## 6 6 6
## 7 7 7
## 8 8 8
## 9 9 9
## 10 10 10
## predictors
## 1 cruise_speed
## 2 engine_type cruise_speed
## 3 engine_type max_speed cruise_speed
## 4 engine_type max_speed cruise_speed stall_speed
## 5 engine_type engine_power max_speed cruise_speed wing_span
## 6 engine_type engine_power max_speed cruise_speed wing_span range
## 7 engine_type engine_power max_speed cruise_speed stall_speed wing_span range
## 8 engine_type engine_power max_speed cruise_speed stall_speed all_eng_roc wing_span range
## 9 engine_type engine_power max_speed cruise_speed stall_speed out_eng_roc takeoff_distance wing_span range
## 10 engine_type engine_power max_speed cruise_speed stall_speed all_eng_roc out_eng_roc takeoff_distance wing_span range
## rsquare adjr predrsq cp aic sbic sbc
## 1 0.8403540 0.8400379 0.8392919 133.86724 14541.31 13101.60 14553.99
## 2 0.8530137 0.8521370 0.8454943 87.36483 14503.42 13061.77 14524.56
## 3 0.8616489 0.8605465 0.8517118 54.91667 14474.72 13033.25 14500.09
## 4 0.8656081 0.8642669 0.8547768 41.12261 14462.00 13020.65 14491.60
## 5 0.8700012 0.8684412 0.8593856 25.59755 14447.15 13006.08 14480.98
## 6 0.8726573 0.8708709 0.8608337 17.00173 14438.69 12997.85 14476.74
## 7 0.8748755 0.8728655 0.8623799 10.15265 14431.78 12991.22 14474.06
## 8 0.8752739 0.8730153 0.8616052 10.56334 14432.16 12991.70 14478.67
## 9 0.8755545 0.8730456 0.8612297 11.44377 14433.02 12992.64 14483.76
## 10 0.8759165 0.8731590 0.8605288 12.00000 14433.54 12993.27 14488.51
## msep fpe apc hsp
## 1 8.416750e+13 166665713962 0.1609105 329386585
## 2 7.764723e+13 154362233774 0.1487362 305077973
## 3 7.323115e+13 145868543974 0.1405515 288300231
## 4 7.127750e+13 142254925506 0.1370690 281169096
## 5 6.908544e+13 138149327661 0.1331126 273067092
## 6 6.780952e+13 135862223961 0.1309083 268561043
## 7 6.676211e+13 134023891075 0.1291365 264943708
## 8 6.668344e+13 134125896389 0.1292343 265163966
## 9 6.666754e+13 134353777160 0.1294533 265635198
## 10 6.660795e+13 134493319232 0.1295873 265933906Using option1(olsrr package): The 7th model is the best model, with CP = 10.15265 and AIC = 14431.78(smallest AIC) and predictors are: engine_type, engine_power, max_speed, cruise_speed, stall_speed, wing_span & range.
#option 1 Continued
rsquare=c((ExecSubsets$metrics)$rsquare)
AdjustedR=c((ExecSubsets$metrics)$adjr)
cp=c((ExecSubsets$metrics)$cp)
AIC=c((ExecSubsets$metrics)$aic)
cbind(rsquare,AdjustedR,cp,AIC)## rsquare AdjustedR cp AIC
## [1,] 0.8403540 0.8400379 133.86724 14541.31
## [2,] 0.8530137 0.8521370 87.36483 14503.42
## [3,] 0.8616489 0.8605465 54.91667 14474.72
## [4,] 0.8656081 0.8642669 41.12261 14462.00
## [5,] 0.8700012 0.8684412 25.59755 14447.15
## [6,] 0.8726573 0.8708709 17.00173 14438.69
## [7,] 0.8748755 0.8728655 10.15265 14431.78
## [8,] 0.8752739 0.8730153 10.56334 14432.16
## [9,] 0.8755545 0.8730456 11.44377 14433.02
## [10,] 0.8759165 0.8731590 12.00000 14433.54#option 2: using leaps package
best.subsetFullModel=regsubsets(price ~ engine_type + engine_power + max_speed + cruise_speed +stall_speed +
all_eng_roc + out_eng_roc + takeoff_distance + wing_span + range, data= AircraftData, nv=10 )
summary(best.subsetFullModel)## Subset selection object
## Call: regsubsets.formula(price ~ engine_type + engine_power + max_speed +
## cruise_speed + stall_speed + all_eng_roc + out_eng_roc +
## takeoff_distance + wing_span + range, data = AircraftData,
## nv = 10)
## 11 Variables (and intercept)
## Forced in Forced out
## engine_typePiston FALSE FALSE
## engine_typePropjet FALSE FALSE
## engine_power FALSE FALSE
## max_speed FALSE FALSE
## cruise_speed FALSE FALSE
## stall_speed FALSE FALSE
## all_eng_roc FALSE FALSE
## out_eng_roc FALSE FALSE
## takeoff_distance FALSE FALSE
## wing_span FALSE FALSE
## range FALSE FALSE
## 1 subsets of each size up to 10
## Selection Algorithm: exhaustive
## engine_typePiston engine_typePropjet engine_power max_speed
## 1 ( 1 ) " " " " " " " "
## 2 ( 1 ) "*" " " " " " "
## 3 ( 1 ) "*" " " " " "*"
## 4 ( 1 ) "*" " " " " "*"
## 5 ( 1 ) "*" " " "*" "*"
## 6 ( 1 ) "*" "*" "*" "*"
## 7 ( 1 ) "*" "*" "*" "*"
## 8 ( 1 ) "*" "*" "*" "*"
## 9 ( 1 ) "*" "*" "*" "*"
## 10 ( 1 ) "*" "*" "*" "*"
## cruise_speed stall_speed all_eng_roc out_eng_roc takeoff_distance
## 1 ( 1 ) "*" " " " " " " " "
## 2 ( 1 ) "*" " " " " " " " "
## 3 ( 1 ) "*" " " " " " " " "
## 4 ( 1 ) "*" "*" " " " " " "
## 5 ( 1 ) "*" " " " " " " " "
## 6 ( 1 ) "*" " " " " " " " "
## 7 ( 1 ) "*" " " " " " " " "
## 8 ( 1 ) "*" "*" " " " " " "
## 9 ( 1 ) "*" "*" "*" " " " "
## 10 ( 1 ) "*" "*" " " "*" "*"
## wing_span range
## 1 ( 1 ) " " " "
## 2 ( 1 ) " " " "
## 3 ( 1 ) " " " "
## 4 ( 1 ) " " " "
## 5 ( 1 ) "*" " "
## 6 ( 1 ) "*" " "
## 7 ( 1 ) "*" "*"
## 8 ( 1 ) "*" "*"
## 9 ( 1 ) "*" "*"
## 10 ( 1 ) "*" "*"reg.summary=summary(best.subsetFullModel)
rsquare=c(reg.summary$rsq)
cp=c(reg.summary$cp)
AdjustedR=c(reg.summary$adjr2)
RSS=c(reg.summary$rss)
BIC=c(reg.summary$bic)
cbind(rsquare,cp,BIC,RSS,AdjustedR)## rsquare cp BIC RSS AdjustedR
## [1,] 0.8403540 133.86724 -917.7849 8.383547e+13 0.8400379
## [2,] 0.8525548 87.19535 -951.8640 7.742844e+13 0.8519697
## [3,] 0.8614890 53.55476 -977.3263 7.273681e+13 0.8606629
## [4,] 0.8646943 42.76774 -982.9685 7.105356e+13 0.8636162
## [5,] 0.8667015 36.76087 -984.3171 6.999956e+13 0.8653711
## [6,] 0.8700012 25.59755 -990.7969 6.826677e+13 0.8684412
## [7,] 0.8726573 17.00173 -995.0346 6.687197e+13 0.8708709
## [8,] 0.8748755 10.15265 -997.7155 6.570710e+13 0.8728655
## [9,] 0.8752739 10.56334 -993.1039 6.549789e+13 0.8730153
## [10,] 0.8755545 11.44377 -988.0174 6.535051e+13 0.8730456Using the option2(with leaps package): The model with AdjustedR2 = 0.8728655, Mallow’sCP = 10.15265, BIC = -997.7155 is the best predictor and predictors are: engine_type, engine_power, max_speed, cruise_speed, stall_speed, wing_span & range.
Thus we can conclude that our best additive model as: price ~ engine_type + engine_power + max_speed + cruise_speed + stall_speed + wing_span + range
In the next step, we will build interaction models based on best additive model.
interactPriceFull = lm(price ~ (factor(engine_type) + engine_power + max_speed +
cruise_speed + stall_speed + wing_span + range)^2, data = AircraftData)
summary(interactPriceFull)##
## Call:
## lm(formula = price ~ (factor(engine_type) + engine_power + max_speed +
## cruise_speed + stall_speed + wing_span + range)^2, data = AircraftData)
##
## Residuals:
## Min 1Q Median 3Q Max
## -973549 -180573 -26201 139343 853490
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 4.256e+06 1.302e+06 3.268 0.001164
## factor(engine_type)Piston -3.258e+06 9.645e+05 -3.378 0.000791
## factor(engine_type)Propjet -1.990e+06 8.984e+05 -2.215 0.027236
## engine_power 1.847e+03 3.102e+02 5.952 5.17e-09
## max_speed -1.218e+04 5.504e+03 -2.213 0.027346
## cruise_speed 7.187e+03 4.248e+03 1.692 0.091361
## stall_speed -2.160e+04 1.887e+04 -1.145 0.252856
## wing_span -8.295e+03 2.848e+03 -2.913 0.003752
## range 2.116e+03 6.499e+02 3.256 0.001212
## factor(engine_type)Piston:engine_power -1.317e+03 3.456e+02 -3.813 0.000156
## factor(engine_type)Propjet:engine_power -2.442e+03 3.324e+02 -7.348 8.93e-13
## factor(engine_type)Piston:max_speed 4.888e+03 2.405e+03 2.032 0.042677
## factor(engine_type)Propjet:max_speed 5.496e+03 1.766e+03 3.111 0.001976
## factor(engine_type)Piston:cruise_speed 7.405e+02 2.432e+03 0.304 0.760915
## factor(engine_type)Propjet:cruise_speed -4.113e+03 1.839e+03 -2.236 0.025807
## factor(engine_type)Piston:stall_speed 4.935e+03 9.977e+03 0.495 0.621117
## factor(engine_type)Propjet:stall_speed 2.555e+03 9.092e+03 0.281 0.778792
## factor(engine_type)Piston:wing_span 8.378e+03 2.152e+03 3.893 0.000113
## factor(engine_type)Propjet:wing_span 8.132e+03 1.868e+03 4.353 1.65e-05
## factor(engine_type)Piston:range -1.349e+03 3.530e+02 -3.822 0.000150
## factor(engine_type)Propjet:range -6.499e+02 2.909e+02 -2.234 0.025943
## engine_power:max_speed -2.107e+00 5.115e-01 -4.119 4.49e-05
## engine_power:cruise_speed -3.480e+00 7.785e-01 -4.471 9.79e-06
## engine_power:stall_speed 9.480e-03 1.875e+00 0.005 0.995967
## engine_power:wing_span 1.013e+00 3.409e-01 2.971 0.003119
## engine_power:range -2.007e-02 3.182e-02 -0.631 0.528524
## max_speed:cruise_speed 1.565e+01 8.910e+00 1.756 0.079692
## max_speed:stall_speed 2.958e+00 1.872e+01 0.158 0.874498
## max_speed:wing_span 2.484e+01 9.414e+00 2.638 0.008613
## max_speed:range -1.977e+00 1.340e+00 -1.475 0.140817
## cruise_speed:stall_speed 7.221e+01 3.182e+01 2.269 0.023694
## cruise_speed:wing_span -1.060e+01 1.153e+01 -0.920 0.358169
## cruise_speed:range -1.439e+00 1.469e+00 -0.979 0.327940
## stall_speed:wing_span -1.562e+01 2.883e+01 -0.542 0.588139
## stall_speed:range 8.615e-01 5.092e+00 0.169 0.865721
## wing_span:range -6.797e-01 5.239e-01 -1.297 0.195189
##
## (Intercept) **
## factor(engine_type)Piston ***
## factor(engine_type)Propjet *
## engine_power ***
## max_speed *
## cruise_speed .
## stall_speed
## wing_span **
## range **
## factor(engine_type)Piston:engine_power ***
## factor(engine_type)Propjet:engine_power ***
## factor(engine_type)Piston:max_speed *
## factor(engine_type)Propjet:max_speed **
## factor(engine_type)Piston:cruise_speed
## factor(engine_type)Propjet:cruise_speed *
## factor(engine_type)Piston:stall_speed
## factor(engine_type)Propjet:stall_speed
## factor(engine_type)Piston:wing_span ***
## factor(engine_type)Propjet:wing_span ***
## factor(engine_type)Piston:range ***
## factor(engine_type)Propjet:range *
## engine_power:max_speed ***
## engine_power:cruise_speed ***
## engine_power:stall_speed
## engine_power:wing_span **
## engine_power:range
## max_speed:cruise_speed .
## max_speed:stall_speed
## max_speed:wing_span **
## max_speed:range
## cruise_speed:stall_speed *
## cruise_speed:wing_span
## cruise_speed:range
## stall_speed:wing_span
## stall_speed:range
## wing_span:range
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 282100 on 471 degrees of freedom
## Multiple R-squared: 0.9286, Adjusted R-squared: 0.9233
## F-statistic: 175.1 on 35 and 471 DF, p-value: < 2.2e-16#insignificant terms are removed and new model is created
interacModel = lm(price ~ factor(engine_type) + engine_power + max_speed +
cruise_speed + stall_speed + wing_span + range +
factor(engine_type) * engine_power + factor(engine_type) * max_speed +
factor(engine_type) * cruise_speed + factor(engine_type) * wing_span +
engine_power * max_speed + engine_power * cruise_speed + engine_power * wing_span +
max_speed * wing_span + cruise_speed * stall_speed, data = AircraftData)
summary(interacModel)##
## Call:
## lm(formula = price ~ factor(engine_type) + engine_power + max_speed +
## cruise_speed + stall_speed + wing_span + range + factor(engine_type) *
## engine_power + factor(engine_type) * max_speed + factor(engine_type) *
## cruise_speed + factor(engine_type) * wing_span + engine_power *
## max_speed + engine_power * cruise_speed + engine_power *
## wing_span + max_speed * wing_span + cruise_speed * stall_speed,
## data = AircraftData)
##
## Residuals:
## Min 1Q Median 3Q Max
## -962876 -184373 -32016 139816 837474
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 4.205e+06 1.079e+06 3.898 0.000111
## factor(engine_type)Piston -3.179e+06 7.852e+05 -4.048 6.01e-05
## factor(engine_type)Propjet -1.889e+06 6.321e+05 -2.989 0.002945
## engine_power 1.726e+03 2.151e+02 8.022 7.86e-15
## max_speed -2.126e+03 2.212e+03 -0.961 0.336890
## cruise_speed 3.563e+03 1.572e+03 2.267 0.023841
## stall_speed -2.058e+04 3.907e+03 -5.266 2.10e-07
## wing_span -6.681e+03 2.198e+03 -3.040 0.002493
## range 7.327e+01 4.460e+01 1.643 0.101044
## factor(engine_type)Piston:engine_power -9.805e+02 2.683e+02 -3.655 0.000286
## factor(engine_type)Propjet:engine_power -2.217e+03 2.761e+02 -8.029 7.48e-15
## factor(engine_type)Piston:max_speed 1.097e+03 1.634e+03 0.671 0.502226
## factor(engine_type)Propjet:max_speed 4.679e+03 1.149e+03 4.074 5.40e-05
## factor(engine_type)Piston:cruise_speed 2.308e+03 1.984e+03 1.163 0.245364
## factor(engine_type)Propjet:cruise_speed -3.238e+03 1.443e+03 -2.244 0.025284
## factor(engine_type)Piston:wing_span 6.354e+03 1.488e+03 4.272 2.34e-05
## factor(engine_type)Propjet:wing_span 6.060e+03 1.267e+03 4.784 2.28e-06
## engine_power:max_speed -1.553e+00 2.889e-01 -5.376 1.19e-07
## engine_power:cruise_speed -2.895e+00 3.857e-01 -7.506 2.95e-13
## engine_power:wing_span 4.083e-01 1.060e-01 3.851 0.000133
## max_speed:wing_span 1.048e+01 4.561e+00 2.298 0.022004
## cruise_speed:stall_speed 6.309e+01 1.275e+01 4.949 1.03e-06
##
## (Intercept) ***
## factor(engine_type)Piston ***
## factor(engine_type)Propjet **
## engine_power ***
## max_speed
## cruise_speed *
## stall_speed ***
## wing_span **
## range
## factor(engine_type)Piston:engine_power ***
## factor(engine_type)Propjet:engine_power ***
## factor(engine_type)Piston:max_speed
## factor(engine_type)Propjet:max_speed ***
## factor(engine_type)Piston:cruise_speed
## factor(engine_type)Propjet:cruise_speed *
## factor(engine_type)Piston:wing_span ***
## factor(engine_type)Propjet:wing_span ***
## engine_power:max_speed ***
## engine_power:cruise_speed ***
## engine_power:wing_span ***
## max_speed:wing_span *
## cruise_speed:stall_speed ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 284300 on 485 degrees of freedom
## Multiple R-squared: 0.9253, Adjusted R-squared: 0.9221
## F-statistic: 286.2 on 21 and 485 DF, p-value: < 2.2e-16#insignicant terms removed from the above interaction model: range
bestInteracModel = lm(price ~ factor(engine_type) + engine_power + max_speed +
cruise_speed + stall_speed + wing_span +
factor(engine_type) * engine_power + factor(engine_type) * max_speed +
factor(engine_type) * cruise_speed + factor(engine_type) * wing_span +
engine_power * max_speed + engine_power * cruise_speed + engine_power * wing_span +
max_speed * wing_span + cruise_speed * stall_speed, data = AircraftData)
summary(bestInteracModel)##
## Call:
## lm(formula = price ~ factor(engine_type) + engine_power + max_speed +
## cruise_speed + stall_speed + wing_span + factor(engine_type) *
## engine_power + factor(engine_type) * max_speed + factor(engine_type) *
## cruise_speed + factor(engine_type) * wing_span + engine_power *
## max_speed + engine_power * cruise_speed + engine_power *
## wing_span + max_speed * wing_span + cruise_speed * stall_speed,
## data = AircraftData)
##
## Residuals:
## Min 1Q Median 3Q Max
## -962455 -183993 -29856 147763 842716
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 4.096e+06 1.079e+06 3.798 0.000165
## factor(engine_type)Piston -3.084e+06 7.845e+05 -3.931 9.68e-05
## factor(engine_type)Propjet -1.852e+06 6.328e+05 -2.927 0.003585
## engine_power 1.733e+03 2.154e+02 8.045 6.66e-15
## max_speed -2.131e+03 2.216e+03 -0.962 0.336611
## cruise_speed 3.856e+03 1.564e+03 2.465 0.014048
## stall_speed -1.973e+04 3.880e+03 -5.086 5.24e-07
## wing_span -6.532e+03 2.200e+03 -2.970 0.003128
## factor(engine_type)Piston:engine_power -1.041e+03 2.662e+02 -3.912 0.000105
## factor(engine_type)Propjet:engine_power -2.191e+03 2.762e+02 -7.935 1.47e-14
## factor(engine_type)Piston:max_speed 1.254e+03 1.634e+03 0.768 0.443107
## factor(engine_type)Propjet:max_speed 4.892e+03 1.143e+03 4.279 2.27e-05
## factor(engine_type)Piston:cruise_speed 2.194e+03 1.987e+03 1.105 0.269874
## factor(engine_type)Propjet:cruise_speed -3.253e+03 1.445e+03 -2.251 0.024860
## factor(engine_type)Piston:wing_span 6.155e+03 1.485e+03 4.144 4.02e-05
## factor(engine_type)Propjet:wing_span 5.861e+03 1.263e+03 4.640 4.49e-06
## engine_power:max_speed -1.591e+00 2.885e-01 -5.513 5.74e-08
## engine_power:cruise_speed -2.918e+00 3.861e-01 -7.559 2.04e-13
## engine_power:wing_span 4.369e-01 1.048e-01 4.171 3.60e-05
## max_speed:wing_span 1.071e+01 4.567e+00 2.346 0.019390
## cruise_speed:stall_speed 6.071e+01 1.269e+01 4.785 2.27e-06
##
## (Intercept) ***
## factor(engine_type)Piston ***
## factor(engine_type)Propjet **
## engine_power ***
## max_speed
## cruise_speed *
## stall_speed ***
## wing_span **
## factor(engine_type)Piston:engine_power ***
## factor(engine_type)Propjet:engine_power ***
## factor(engine_type)Piston:max_speed
## factor(engine_type)Propjet:max_speed ***
## factor(engine_type)Piston:cruise_speed
## factor(engine_type)Propjet:cruise_speed *
## factor(engine_type)Piston:wing_span ***
## factor(engine_type)Propjet:wing_span ***
## engine_power:max_speed ***
## engine_power:cruise_speed ***
## engine_power:wing_span ***
## max_speed:wing_span *
## cruise_speed:stall_speed ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 284800 on 486 degrees of freedom
## Multiple R-squared: 0.9249, Adjusted R-squared: 0.9218
## F-statistic: 299.3 on 20 and 486 DF, p-value: < 2.2e-16bestInteracModel => price ~ factor(engine_type) + engine_power +
max_speed + cruise_speed + stall_speed + wing_span + factor(engine_type)
* engine_power + factor(engine_type) * max_speed + factor(engine_type) *
cruise_speed + factor(engine_type) * wing_span +
engine_power * max_speed + engine_power * cruise_speed + engine_power *
wing_span + max_speed * wing_span + cruise_speed * stall_speed
We have to test which model is to be taken as the best interaction model. So we can perform an ANOVA test. H0: the reduced model is the best H1: the full interaction model is the best
anova(bestInteracModel, interactPriceFull)## Analysis of Variance Table
##
## Model 1: price ~ factor(engine_type) + engine_power + max_speed + cruise_speed +
## stall_speed + wing_span + factor(engine_type) * engine_power +
## factor(engine_type) * max_speed + factor(engine_type) * cruise_speed +
## factor(engine_type) * wing_span + engine_power * max_speed +
## engine_power * cruise_speed + engine_power * wing_span +
## max_speed * wing_span + cruise_speed * stall_speed
## Model 2: price ~ (factor(engine_type) + engine_power + max_speed + cruise_speed +
## stall_speed + wing_span + range)^2
## Res.Df RSS Df Sum of Sq F Pr(>F)
## 1 486 3.9428e+13
## 2 471 3.7475e+13 15 1.9531e+12 1.6365 0.06086 .
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1From the above output, we can see that the p-value = 0.06086 > aplha(0.05). So we fail to reject the null hypothesis. There for we can say that the reduced model is the best.
We can check if there is a possibility for higher order for the predictors using the pairs plot.
pairs(~price + factor(engine_type) +engine_power, data = AircraftData , panel = panel.smooth)
pairs(~price + max_speed + cruise_speed + stall_speed, data = AircraftData, panel = panel.smooth)
pairs(~price + wing_span + range ,data = AircraftData, panel = panel.smooth)
pairs(~price + engine_power + max_speed + cruise_speed + stall_speed + wing_span + range ,data = AircraftData, panel = panel.smooth)
From the above output, we can see there might be a possibility for higher orders for the variables: engine_power, max_speed and cruise_speed.
# Checking I(engine_power^2)
higherOrder1 = lm(price ~ factor(engine_type) + engine_power + I(engine_power^2) +
max_speed + cruise_speed + stall_speed + wing_span +
factor(engine_type) * engine_power + factor(engine_type) * max_speed +
factor(engine_type) * cruise_speed + factor(engine_type) * wing_span +
engine_power * max_speed + engine_power * cruise_speed + engine_power * wing_span +
max_speed * wing_span + cruise_speed * stall_speed, data = AircraftData)
summary(higherOrder1)##
## Call:
## lm(formula = price ~ factor(engine_type) + engine_power + I(engine_power^2) +
## max_speed + cruise_speed + stall_speed + wing_span + factor(engine_type) *
## engine_power + factor(engine_type) * max_speed + factor(engine_type) *
## cruise_speed + factor(engine_type) * wing_span + engine_power *
## max_speed + engine_power * cruise_speed + engine_power *
## wing_span + max_speed * wing_span + cruise_speed * stall_speed,
## data = AircraftData)
##
## Residuals:
## Min 1Q Median 3Q Max
## -963496 -186337 -30832 146997 840584
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 3.923e+06 1.140e+06 3.442 0.000626
## factor(engine_type)Piston -2.905e+06 8.709e+05 -3.336 0.000916
## factor(engine_type)Propjet -1.707e+06 7.031e+05 -2.428 0.015544
## engine_power 1.774e+03 2.320e+02 7.644 1.13e-13
## I(engine_power^2) 4.984e-03 1.050e-02 0.475 0.635092
## max_speed -2.120e+03 2.218e+03 -0.956 0.339733
## cruise_speed 4.120e+03 1.662e+03 2.480 0.013486
## stall_speed -1.964e+04 3.888e+03 -5.052 6.21e-07
## wing_span -6.349e+03 2.235e+03 -2.841 0.004691
## factor(engine_type)Piston:engine_power -1.032e+03 2.671e+02 -3.863 0.000127
## factor(engine_type)Propjet:engine_power -2.166e+03 2.817e+02 -7.688 8.37e-14
## factor(engine_type)Piston:max_speed 1.118e+03 1.661e+03 0.673 0.500984
## factor(engine_type)Propjet:max_speed 4.750e+03 1.183e+03 4.017 6.84e-05
## factor(engine_type)Piston:cruise_speed 1.971e+03 2.043e+03 0.965 0.335254
## factor(engine_type)Propjet:cruise_speed -3.426e+03 1.492e+03 -2.297 0.022064
## factor(engine_type)Piston:wing_span 5.939e+03 1.555e+03 3.821 0.000150
## factor(engine_type)Propjet:wing_span 5.673e+03 1.324e+03 4.283 2.22e-05
## engine_power:max_speed -1.649e+00 3.140e-01 -5.252 2.25e-07
## engine_power:cruise_speed -2.968e+00 4.001e-01 -7.418 5.36e-13
## engine_power:wing_span 3.600e-01 1.929e-01 1.866 0.062612
## max_speed:wing_span 1.105e+01 4.624e+00 2.389 0.017288
## cruise_speed:stall_speed 6.003e+01 1.278e+01 4.698 3.42e-06
##
## (Intercept) ***
## factor(engine_type)Piston ***
## factor(engine_type)Propjet *
## engine_power ***
## I(engine_power^2)
## max_speed
## cruise_speed *
## stall_speed ***
## wing_span **
## factor(engine_type)Piston:engine_power ***
## factor(engine_type)Propjet:engine_power ***
## factor(engine_type)Piston:max_speed
## factor(engine_type)Propjet:max_speed ***
## factor(engine_type)Piston:cruise_speed
## factor(engine_type)Propjet:cruise_speed *
## factor(engine_type)Piston:wing_span ***
## factor(engine_type)Propjet:wing_span ***
## engine_power:max_speed ***
## engine_power:cruise_speed ***
## engine_power:wing_span .
## max_speed:wing_span *
## cruise_speed:stall_speed ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 285100 on 485 degrees of freedom
## Multiple R-squared: 0.925, Adjusted R-squared: 0.9217
## F-statistic: 284.6 on 21 and 485 DF, p-value: < 2.2e-16Since I(engine_power^2) is not significant with the p-value, we are moving to next step.
# Checking I(max_speed^2)
higherOrderMaxSpeed2 = lm(price ~ factor(engine_type) + engine_power + I(max_speed^2) +
max_speed + cruise_speed + stall_speed + wing_span +
factor(engine_type) * engine_power + factor(engine_type) * max_speed +
factor(engine_type) * cruise_speed + factor(engine_type) * wing_span +
engine_power * max_speed + engine_power * cruise_speed + engine_power * wing_span +
max_speed * wing_span + cruise_speed * stall_speed, data = AircraftData)
# Checking I(max_speed^3)
higherOrderMaxSpeed3 = lm(price ~ factor(engine_type) + engine_power + I(max_speed^2) + I(max_speed^3) +
max_speed + cruise_speed + stall_speed + wing_span +
factor(engine_type) * engine_power + factor(engine_type) * max_speed +
factor(engine_type) * cruise_speed + factor(engine_type) * wing_span +
engine_power * max_speed + engine_power * cruise_speed + engine_power * wing_span +
max_speed * wing_span + cruise_speed * stall_speed, data = AircraftData)
# Checking I(max_speed^4)
higherOrderMaxSpeed4 = lm(price ~ factor(engine_type) + engine_power + I(max_speed^2) + I(max_speed^3) + I(max_speed^4) +
max_speed + cruise_speed + stall_speed + wing_span +
factor(engine_type) * engine_power + factor(engine_type) * max_speed +
factor(engine_type) * cruise_speed + factor(engine_type) * wing_span +
engine_power * max_speed + engine_power * cruise_speed + engine_power * wing_span +
max_speed * wing_span + cruise_speed * stall_speed, data = AircraftData)
summary(higherOrderMaxSpeed2)##
## Call:
## lm(formula = price ~ factor(engine_type) + engine_power + I(max_speed^2) +
## max_speed + cruise_speed + stall_speed + wing_span + factor(engine_type) *
## engine_power + factor(engine_type) * max_speed + factor(engine_type) *
## cruise_speed + factor(engine_type) * wing_span + engine_power *
## max_speed + engine_power * cruise_speed + engine_power *
## wing_span + max_speed * wing_span + cruise_speed * stall_speed,
## data = AircraftData)
##
## Residuals:
## Min 1Q Median 3Q Max
## -878011 -179243 -33321 135051 877066
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 4.234e+06 1.073e+06 3.945 9.16e-05
## factor(engine_type)Piston -3.288e+06 7.836e+05 -4.196 3.23e-05
## factor(engine_type)Propjet -2.126e+06 6.376e+05 -3.335 0.000918
## engine_power 1.738e+03 2.141e+02 8.118 3.95e-15
## I(max_speed^2) 3.679e+00 1.398e+00 2.632 0.008755
## max_speed -3.740e+03 2.286e+03 -1.636 0.102456
## cruise_speed 3.790e+03 1.555e+03 2.438 0.015147
## stall_speed -1.935e+04 3.860e+03 -5.014 7.49e-07
## wing_span -6.065e+03 2.193e+03 -2.765 0.005905
## factor(engine_type)Piston:engine_power -1.053e+03 2.646e+02 -3.980 7.96e-05
## factor(engine_type)Propjet:engine_power -2.182e+03 2.745e+02 -7.949 1.33e-14
## factor(engine_type)Piston:max_speed 2.477e+03 1.690e+03 1.466 0.143211
## factor(engine_type)Propjet:max_speed 5.720e+03 1.179e+03 4.851 1.66e-06
## factor(engine_type)Piston:cruise_speed 2.373e+03 1.976e+03 1.201 0.230342
## factor(engine_type)Propjet:cruise_speed -2.945e+03 1.441e+03 -2.043 0.041580
## factor(engine_type)Piston:wing_span 6.040e+03 1.477e+03 4.090 5.05e-05
## factor(engine_type)Propjet:wing_span 5.948e+03 1.256e+03 4.736 2.86e-06
## engine_power:max_speed -1.491e+00 2.893e-01 -5.153 3.73e-07
## engine_power:cruise_speed -2.962e+00 3.841e-01 -7.713 7.05e-14
## engine_power:wing_span 4.321e-01 1.041e-01 4.149 3.94e-05
## max_speed:wing_span 8.488e+00 4.617e+00 1.838 0.066636
## cruise_speed:stall_speed 5.985e+01 1.261e+01 4.744 2.76e-06
##
## (Intercept) ***
## factor(engine_type)Piston ***
## factor(engine_type)Propjet ***
## engine_power ***
## I(max_speed^2) **
## max_speed
## cruise_speed *
## stall_speed ***
## wing_span **
## factor(engine_type)Piston:engine_power ***
## factor(engine_type)Propjet:engine_power ***
## factor(engine_type)Piston:max_speed
## factor(engine_type)Propjet:max_speed ***
## factor(engine_type)Piston:cruise_speed
## factor(engine_type)Propjet:cruise_speed *
## factor(engine_type)Piston:wing_span ***
## factor(engine_type)Propjet:wing_span ***
## engine_power:max_speed ***
## engine_power:cruise_speed ***
## engine_power:wing_span ***
## max_speed:wing_span .
## cruise_speed:stall_speed ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 283100 on 485 degrees of freedom
## Multiple R-squared: 0.926, Adjusted R-squared: 0.9228
## F-statistic: 288.9 on 21 and 485 DF, p-value: < 2.2e-16summary(higherOrderMaxSpeed3)##
## Call:
## lm(formula = price ~ factor(engine_type) + engine_power + I(max_speed^2) +
## I(max_speed^3) + max_speed + cruise_speed + stall_speed +
## wing_span + factor(engine_type) * engine_power + factor(engine_type) *
## max_speed + factor(engine_type) * cruise_speed + factor(engine_type) *
## wing_span + engine_power * max_speed + engine_power * cruise_speed +
## engine_power * wing_span + max_speed * wing_span + cruise_speed *
## stall_speed, data = AircraftData)
##
## Residuals:
## Min 1Q Median 3Q Max
## -832249 -176376 -34849 133272 861647
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 5.030e+06 1.104e+06 4.555 6.65e-06
## factor(engine_type)Piston -3.717e+06 7.936e+05 -4.683 3.67e-06
## factor(engine_type)Propjet -2.303e+06 6.365e+05 -3.619 0.000327
## engine_power 1.598e+03 2.186e+02 7.311 1.11e-12
## I(max_speed^2) 2.490e+01 7.806e+00 3.190 0.001515
## I(max_speed^3) -1.708e-02 6.181e-03 -2.763 0.005951
## max_speed -1.065e+04 3.378e+03 -3.153 0.001717
## cruise_speed 3.005e+03 1.570e+03 1.913 0.056313
## stall_speed -1.871e+04 3.840e+03 -4.872 1.50e-06
## wing_span -5.586e+03 2.185e+03 -2.556 0.010890
## factor(engine_type)Piston:engine_power -8.734e+02 2.708e+02 -3.226 0.001342
## factor(engine_type)Propjet:engine_power -2.152e+03 2.729e+02 -7.886 2.10e-14
## factor(engine_type)Piston:max_speed 3.816e+03 1.747e+03 2.185 0.029386
## factor(engine_type)Propjet:max_speed 5.694e+03 1.171e+03 4.862 1.58e-06
## factor(engine_type)Piston:cruise_speed 3.559e+03 2.009e+03 1.772 0.077078
## factor(engine_type)Propjet:cruise_speed -1.834e+03 1.487e+03 -1.233 0.218060
## factor(engine_type)Piston:wing_span 5.581e+03 1.476e+03 3.780 0.000176
## factor(engine_type)Propjet:wing_span 5.698e+03 1.251e+03 4.556 6.62e-06
## engine_power:max_speed -1.453e+00 2.876e-01 -5.051 6.23e-07
## engine_power:cruise_speed -2.725e+00 3.911e-01 -6.968 1.06e-11
## engine_power:wing_span 4.370e-01 1.034e-01 4.224 2.87e-05
## max_speed:wing_span 7.619e+00 4.597e+00 1.657 0.098103
## cruise_speed:stall_speed 5.579e+01 1.262e+01 4.422 1.21e-05
##
## (Intercept) ***
## factor(engine_type)Piston ***
## factor(engine_type)Propjet ***
## engine_power ***
## I(max_speed^2) **
## I(max_speed^3) **
## max_speed **
## cruise_speed .
## stall_speed ***
## wing_span *
## factor(engine_type)Piston:engine_power **
## factor(engine_type)Propjet:engine_power ***
## factor(engine_type)Piston:max_speed *
## factor(engine_type)Propjet:max_speed ***
## factor(engine_type)Piston:cruise_speed .
## factor(engine_type)Propjet:cruise_speed
## factor(engine_type)Piston:wing_span ***
## factor(engine_type)Propjet:wing_span ***
## engine_power:max_speed ***
## engine_power:cruise_speed ***
## engine_power:wing_span ***
## max_speed:wing_span .
## cruise_speed:stall_speed ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 281200 on 484 degrees of freedom
## Multiple R-squared: 0.9271, Adjusted R-squared: 0.9238
## F-statistic: 279.9 on 22 and 484 DF, p-value: < 2.2e-16summary(higherOrderMaxSpeed4)##
## Call:
## lm(formula = price ~ factor(engine_type) + engine_power + I(max_speed^2) +
## I(max_speed^3) + I(max_speed^4) + max_speed + cruise_speed +
## stall_speed + wing_span + factor(engine_type) * engine_power +
## factor(engine_type) * max_speed + factor(engine_type) * cruise_speed +
## factor(engine_type) * wing_span + engine_power * max_speed +
## engine_power * cruise_speed + engine_power * wing_span +
## max_speed * wing_span + cruise_speed * stall_speed, data = AircraftData)
##
## Residuals:
## Min 1Q Median 3Q Max
## -830583 -175564 -35477 134014 863405
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 4.982e+06 1.116e+06 4.465 1.00e-05
## factor(engine_type)Piston -3.589e+06 8.919e+05 -4.024 6.64e-05
## factor(engine_type)Propjet -2.196e+06 7.225e+05 -3.040 0.002497
## engine_power 1.596e+03 2.189e+02 7.291 1.27e-12
## I(max_speed^2) 3.368e+01 2.901e+01 1.161 0.246156
## I(max_speed^3) -3.651e-02 6.216e-02 -0.587 0.557173
## I(max_speed^4) 1.329e-05 4.230e-05 0.314 0.753434
## max_speed -1.190e+04 5.213e+03 -2.282 0.022915
## cruise_speed 3.080e+03 1.590e+03 1.937 0.053332
## stall_speed -1.874e+04 3.845e+03 -4.874 1.49e-06
## wing_span -5.431e+03 2.243e+03 -2.421 0.015829
## factor(engine_type)Piston:engine_power -8.605e+02 2.741e+02 -3.139 0.001798
## factor(engine_type)Propjet:engine_power -2.143e+03 2.747e+02 -7.798 3.89e-14
## factor(engine_type)Piston:max_speed 3.599e+03 1.880e+03 1.915 0.056143
## factor(engine_type)Propjet:max_speed 5.532e+03 1.281e+03 4.319 1.90e-05
## factor(engine_type)Piston:cruise_speed 3.505e+03 2.018e+03 1.737 0.083014
## factor(engine_type)Propjet:cruise_speed -1.948e+03 1.532e+03 -1.272 0.204152
## factor(engine_type)Piston:wing_span 5.442e+03 1.543e+03 3.527 0.000461
## factor(engine_type)Propjet:wing_span 5.594e+03 1.295e+03 4.320 1.90e-05
## engine_power:max_speed -1.437e+00 2.923e-01 -4.916 1.21e-06
## engine_power:cruise_speed -2.733e+00 3.924e-01 -6.966 1.07e-11
## engine_power:wing_span 4.338e-01 1.040e-01 4.171 3.60e-05
## max_speed:wing_span 7.428e+00 4.641e+00 1.600 0.110165
## cruise_speed:stall_speed 5.586e+01 1.263e+01 4.423 1.20e-05
##
## (Intercept) ***
## factor(engine_type)Piston ***
## factor(engine_type)Propjet **
## engine_power ***
## I(max_speed^2)
## I(max_speed^3)
## I(max_speed^4)
## max_speed *
## cruise_speed .
## stall_speed ***
## wing_span *
## factor(engine_type)Piston:engine_power **
## factor(engine_type)Propjet:engine_power ***
## factor(engine_type)Piston:max_speed .
## factor(engine_type)Propjet:max_speed ***
## factor(engine_type)Piston:cruise_speed .
## factor(engine_type)Propjet:cruise_speed
## factor(engine_type)Piston:wing_span ***
## factor(engine_type)Propjet:wing_span ***
## engine_power:max_speed ***
## engine_power:cruise_speed ***
## engine_power:wing_span ***
## max_speed:wing_span
## cruise_speed:stall_speed ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 281500 on 483 degrees of freedom
## Multiple R-squared: 0.9271, Adjusted R-squared: 0.9237
## F-statistic: 267.2 on 23 and 483 DF, p-value: < 2.2e-16The quartic term is not significant, So we proceed with the model with cubic term.
# Checking I(cruise_speed^2)
higherOrderCruiseSpeed2 = lm(price ~ factor(engine_type) + engine_power + I(cruise_speed^2) +
max_speed + cruise_speed + stall_speed + wing_span +
factor(engine_type) * engine_power + factor(engine_type) * max_speed +
factor(engine_type) * cruise_speed + factor(engine_type) * wing_span +
engine_power * max_speed + engine_power * cruise_speed + engine_power * wing_span +
max_speed * wing_span + cruise_speed * stall_speed, data = AircraftData)
# Checking I(cruise_speed^3)
higherOrderCruiseSpeed3 = lm(price ~ factor(engine_type) + engine_power + I(cruise_speed^2) + I(cruise_speed^3) +
max_speed + cruise_speed + stall_speed + wing_span +
factor(engine_type) * engine_power + factor(engine_type) * max_speed +
factor(engine_type) * cruise_speed + factor(engine_type) * wing_span +
engine_power * max_speed + engine_power * cruise_speed + engine_power * wing_span +
max_speed * wing_span + cruise_speed * stall_speed, data = AircraftData)
summary(higherOrderCruiseSpeed2)##
## Call:
## lm(formula = price ~ factor(engine_type) + engine_power + I(cruise_speed^2) +
## max_speed + cruise_speed + stall_speed + wing_span + factor(engine_type) *
## engine_power + factor(engine_type) * max_speed + factor(engine_type) *
## cruise_speed + factor(engine_type) * wing_span + engine_power *
## max_speed + engine_power * cruise_speed + engine_power *
## wing_span + max_speed * wing_span + cruise_speed * stall_speed,
## data = AircraftData)
##
## Residuals:
## Min 1Q Median 3Q Max
## -910848 -177986 -30919 135247 866456
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 5.823e+06 1.127e+06 5.168 3.47e-07
## factor(engine_type)Piston -4.389e+06 8.234e+05 -5.330 1.51e-07
## factor(engine_type)Propjet -2.755e+06 6.530e+05 -4.219 2.93e-05
## engine_power 1.700e+03 2.115e+02 8.037 7.09e-15
## I(cruise_speed^2) 2.100e+01 4.707e+00 4.461 1.02e-05
## max_speed -3.170e+03 2.187e+03 -1.450 0.147823
## cruise_speed -7.506e+03 2.974e+03 -2.524 0.011918
## stall_speed -1.736e+04 3.844e+03 -4.516 7.92e-06
## wing_span -6.491e+03 2.158e+03 -3.008 0.002768
## factor(engine_type)Piston:engine_power -9.810e+02 2.615e+02 -3.751 0.000197
## factor(engine_type)Propjet:engine_power -2.285e+03 2.718e+02 -8.406 4.74e-16
## factor(engine_type)Piston:max_speed 1.571e+03 1.605e+03 0.979 0.328265
## factor(engine_type)Propjet:max_speed 5.697e+03 1.136e+03 5.015 7.46e-07
## factor(engine_type)Piston:cruise_speed 8.318e+03 2.384e+03 3.489 0.000529
## factor(engine_type)Propjet:cruise_speed -1.528e+02 1.579e+03 -0.097 0.922938
## factor(engine_type)Piston:wing_span 5.975e+03 1.458e+03 4.099 4.86e-05
## factor(engine_type)Propjet:wing_span 5.968e+03 1.239e+03 4.815 1.97e-06
## engine_power:max_speed -1.270e+00 2.921e-01 -4.348 1.68e-05
## engine_power:cruise_speed -3.213e+00 3.845e-01 -8.357 6.85e-16
## engine_power:wing_span 4.307e-01 1.028e-01 4.190 3.31e-05
## max_speed:wing_span 1.064e+01 4.481e+00 2.374 0.017984
## cruise_speed:stall_speed 4.920e+01 1.271e+01 3.871 0.000123
##
## (Intercept) ***
## factor(engine_type)Piston ***
## factor(engine_type)Propjet ***
## engine_power ***
## I(cruise_speed^2) ***
## max_speed
## cruise_speed *
## stall_speed ***
## wing_span **
## factor(engine_type)Piston:engine_power ***
## factor(engine_type)Propjet:engine_power ***
## factor(engine_type)Piston:max_speed
## factor(engine_type)Propjet:max_speed ***
## factor(engine_type)Piston:cruise_speed ***
## factor(engine_type)Propjet:cruise_speed
## factor(engine_type)Piston:wing_span ***
## factor(engine_type)Propjet:wing_span ***
## engine_power:max_speed ***
## engine_power:cruise_speed ***
## engine_power:wing_span ***
## max_speed:wing_span *
## cruise_speed:stall_speed ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 279400 on 485 degrees of freedom
## Multiple R-squared: 0.9279, Adjusted R-squared: 0.9248
## F-statistic: 297.1 on 21 and 485 DF, p-value: < 2.2e-16summary(higherOrderCruiseSpeed3)##
## Call:
## lm(formula = price ~ factor(engine_type) + engine_power + I(cruise_speed^2) +
## I(cruise_speed^3) + max_speed + cruise_speed + stall_speed +
## wing_span + factor(engine_type) * engine_power + factor(engine_type) *
## max_speed + factor(engine_type) * cruise_speed + factor(engine_type) *
## wing_span + engine_power * max_speed + engine_power * cruise_speed +
## engine_power * wing_span + max_speed * wing_span + cruise_speed *
## stall_speed, data = AircraftData)
##
## Residuals:
## Min 1Q Median 3Q Max
## -908287 -178593 -32938 137168 864486
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 5.777e+06 1.145e+06 5.043 6.48e-07
## factor(engine_type)Piston -4.393e+06 8.244e+05 -5.329 1.52e-07
## factor(engine_type)Propjet -2.774e+06 6.589e+05 -4.211 3.04e-05
## engine_power 1.697e+03 2.121e+02 7.999 9.34e-15
## I(cruise_speed^2) 1.764e+01 1.522e+01 1.160 0.246775
## I(cruise_speed^3) 4.256e-03 1.837e-02 0.232 0.816899
## max_speed -3.217e+03 2.198e+03 -1.463 0.143981
## cruise_speed -6.697e+03 4.586e+03 -1.460 0.144831
## stall_speed -1.734e+04 3.849e+03 -4.504 8.37e-06
## wing_span -6.479e+03 2.161e+03 -2.999 0.002850
## factor(engine_type)Piston:engine_power -9.842e+02 2.621e+02 -3.755 0.000195
## factor(engine_type)Propjet:engine_power -2.279e+03 2.732e+02 -8.340 7.80e-16
## factor(engine_type)Piston:max_speed 1.672e+03 1.665e+03 1.004 0.315802
## factor(engine_type)Propjet:max_speed 5.729e+03 1.145e+03 5.002 7.96e-07
## factor(engine_type)Piston:cruise_speed 8.179e+03 2.460e+03 3.325 0.000951
## factor(engine_type)Propjet:cruise_speed -1.095e+02 1.592e+03 -0.069 0.945166
## factor(engine_type)Piston:wing_span 5.977e+03 1.459e+03 4.096 4.92e-05
## factor(engine_type)Propjet:wing_span 5.953e+03 1.242e+03 4.792 2.19e-06
## engine_power:max_speed -1.251e+00 3.028e-01 -4.132 4.23e-05
## engine_power:cruise_speed -3.228e+00 3.901e-01 -8.274 1.27e-15
## engine_power:wing_span 4.303e-01 1.029e-01 4.182 3.43e-05
## max_speed:wing_span 1.062e+01 4.485e+00 2.369 0.018240
## cruise_speed:stall_speed 4.917e+01 1.273e+01 3.864 0.000127
##
## (Intercept) ***
## factor(engine_type)Piston ***
## factor(engine_type)Propjet ***
## engine_power ***
## I(cruise_speed^2)
## I(cruise_speed^3)
## max_speed
## cruise_speed
## stall_speed ***
## wing_span **
## factor(engine_type)Piston:engine_power ***
## factor(engine_type)Propjet:engine_power ***
## factor(engine_type)Piston:max_speed
## factor(engine_type)Propjet:max_speed ***
## factor(engine_type)Piston:cruise_speed ***
## factor(engine_type)Propjet:cruise_speed
## factor(engine_type)Piston:wing_span ***
## factor(engine_type)Propjet:wing_span ***
## engine_power:max_speed ***
## engine_power:cruise_speed ***
## engine_power:wing_span ***
## max_speed:wing_span *
## cruise_speed:stall_speed ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 279700 on 484 degrees of freedom
## Multiple R-squared: 0.9279, Adjusted R-squared: 0.9246
## F-statistic: 283.1 on 22 and 484 DF, p-value: < 2.2e-16The cubic term for cruise_speed is not significant, So we proceed with the model with quadratic term.
summary(higherOrderMaxSpeed3)$adj.r.squared## [1] 0.9238114sigma(higherOrderMaxSpeed3)## [1] 281193.1summary(higherOrderCruiseSpeed2)$adj.r.squared## [1] 0.9247535sigma(higherOrderCruiseSpeed2)## [1] 279449.2Based on the ouput above: Adjusted r square is higher for higherOrderCruiseSpeed2 model and RSE is lower,So we could say that the higherOrderMaxSpeed3 is the best model.
So from all the above models we can conclude our best model as:
price ~ factor(engine_type) + engine_power + I(cruise_speed^2) + max_speed + cruise_speed + stall_speed + wing_span + factor(engine_type) * engine_power + factor(engine_type) * max_speed + factor(engine_type) * cruise_speed + factor(engine_type) * wing_span + engine_power * max_speed + engine_power * cruise_speed + engine_power * wing_span + max_speed * wing_span + cruise_speed * stall_speed
\(\widehat{\text{price}}\) = 5823241.56 - 4388887.92Xengine_typePiston - 2755059.51Xengine_typePropjet + 21Xcruise_speed2 + 1699.67Xengine_power - 3169.59Xmax_speed - 7505.68Xcruise_speed - 17359.94Xstall_speed - 6490.75Xwing_span - 981X(engine_type)Piston * Xengine_power - 2284.54X(engine_type)Propjet * Xengine_power + 1570.64X(engine_type)Piston * Xmax_speed + 5697.49X(engine_type)Propjet * Xmax_speed + 8317.65X(engine_type)Piston * Xcruise_speed - 152.84X(engine_type)Propjet * Xcruise_speed + 5975.35X(engine_type)Piston * Xwing_span + 5967.59X(engine_type)Propjet * Xwing_span - 1.27Xengine_power * Xmax_speed - 3.21Xengine_power * Xcruise_speed + 0.43Xengine_power * Xwing_span + 10.64Xmax_speed * Xwing_span + 49.2Xcruise_speed * Xstall_speed
bestModel = lm(price ~ factor(engine_type) + engine_power + I(cruise_speed^2) +
max_speed + cruise_speed + stall_speed + wing_span +
factor(engine_type) * engine_power + factor(engine_type) * max_speed +
factor(engine_type) * cruise_speed + factor(engine_type) * wing_span +
engine_power * max_speed + engine_power * cruise_speed + engine_power * wing_span +
max_speed * wing_span + cruise_speed * stall_speed, data = AircraftData)\(\beta_0\)= 5823241.56
The price of an aircraft with jet engine type is $5.8 million when all
the other predictors are zero.(This is not valid in real life
situations)
\(\beta_~(enginetype)Piston\)= - 4388887.92 Aircraft with a Piston engine are, on average, $4.4 million cheaper than the aircrafts with Jet engine type holding all other variables constant.
\(\beta_~(enginetype)Propjet\)= -2755059.51 The price of an aircraft with a Propjet engine is $2.8 million lesser than the jet engine type aircrafts. Still cheaper, but not as much as a piston plane.
\(\beta~cruise_speed^2\)= 21 The squared term suggests a nonlinear relationship between cruise speed and price. For a 1 unit increase in cruise_speed, the price increases by 21*\(cruisespeed^2\), indicating that price rises more steeply at higher cruise speeds.
\(\beta~enginepower\)= 1699.67
A 1-horsepower increase in engine_power price jump by $1,699.67,
reflecting the value of greater power.we know —stronger engines aren’t
cheap.
\(\beta~maxspeed\)= -3169.59
For every 1 knot increase of max speed, the price drops by $3,170.
usually Faster planes should cost more but here this effect is tangled
up with other factors we will get to—like interactions.
\(\beta~cruisespeed\)=
-7505.68
A 1-knot increase in cruise speed decreases price by
$7,505.68.##########################checkkkkk
\(\beta~stallspeed\)=
-17359.94
If the stall speed goes up by a one knot, the price drops by
$17,360.
\(\beta~wingspan\)= -6490.75
A 1-inch increase in wingspan decreases price by $6,491.
\(\beta_~(enginetype)Piston*enginepower\)=
−981
For Piston engines, each additional horsepower reduces the price
increase from engine_power by $981 compared to the reference type. Net
effect for Piston: 1,699.67 − 981 = 718.67 dollars per
horsepower.
\(\beta_~(enginetype)Propjet*enginepower\)= −2,284.54 For Propjet engines, each horsepower reduces the price increase by $2,284.54.
\(\beta_~(enginetype)Piston*maxspeed\)=
1,570.64
For Piston engines, a 1-knot increase in max speed increases price by an
additional $1,570.64 over the reference. Net effect: −3,169.59 +
1,570.64= −1,598.95 dollars per unit.With piston engines, an knot
increase of max speed cuts the price drop to $1,599.
\(\beta_~(enginetype)Propjet*maxspeed\)=
5,697.49
For Propjets, a 1-knot increase in max speed adds $5,697.49 to the price
of the aircraft. Net effect:
−3,169.59+5,697.49 = 2,527.90, indicating higher max speed boosts price
for Propjets.
\(\beta_~(enginetype)Piston*cruisespeed\)=
8,317.65
For Piston engines, a 1-knot increase in cruise speed adds $8,317.65.
Net effect: −7,505.68 + 8,317.65= 811.97 dollars per unit.Here, a knot
increase of cruise speed adds $812 to the price for aircrafts with
piston engine.
\(\beta_~(enginetype)Propjet*cruisespeed\)= −
152.84
For Propjets, a 1-knot increase reduces the cruise speed effect by
$152.84. Net effect:
−7,505.68 − 152.84= −7,658.52.For propjets, cruise speed still drags the
price down, by $7,659 per knot.
\(\beta_~(enginetype)Piston*wingspan\)=
5,975.35
For Piston engines, a 1-inch increase in wingspan adds $5,975.35 to the
aircraft price.
Net effect: −6,490.75+5,975.35= −515.40.piston planes soften the
wingspan price drop to about $515 per inch.
\(\beta_~(enginetype)Propjet*wingspan\)=
5,967.59
For Propjets, a 1-inch increase in wingspan increases the aircraft price
by $5,967.59.
Net effect: −6,490.75 + 5,967.59=−523.16.Propjet planes soften the
wingspan price drop to about $515 per inch.
\(\beta_~enginepower*maxspeed\)=
-1.27
For a 1-knot increase in max speed, the effect of engine power on price
decreases by $1.27 per horsepower.
\(\beta_~enginepower*cruisespeed\)=
-3.21
For a 1-knot increase in cruise speed, the effect of engine power on
aircraft price decreases by $3.21 per horsepower.
\(\beta_~enginepower*wingspan\)=
0.43
For a 1-inch increase in wingspan, the effect of engine power on
aircraft price increases by $0.43 per horsepower.
\(\beta_~maxspeed*wingspan\)=
10.64
For a 1-inch increase in wingspan, the effect of max speed on aircraft
price increases by $10.64 per knot.
\(\beta_~cruisespeed*stallspeed\)=
49.2
For a 1-knot increase in stall speed, the effect of cruise speed on
price of the aircraft increases by $49.2 per knot.
conf_intervals <- confint(bestModel, level = 0.95)
print(conf_intervals)## 2.5 % 97.5 %
## (Intercept) 3.609132e+06 8.037351e+06
## factor(engine_type)Piston -6.006723e+06 -2.771053e+06
## factor(engine_type)Propjet -4.038185e+06 -1.471934e+06
## engine_power 1.284119e+03 2.115226e+03
## I(cruise_speed^2) 1.174739e+01 3.024338e+01
## max_speed -7.465888e+03 1.126718e+03
## cruise_speed -1.334846e+04 -1.662894e+03
## stall_speed -2.491285e+04 -9.807029e+03
## wing_span -1.073089e+04 -2.250609e+03
## factor(engine_type)Piston:engine_power -1.494853e+03 -4.671547e+02
## factor(engine_type)Propjet:engine_power -2.818517e+03 -1.750558e+03
## factor(engine_type)Piston:max_speed -1.582947e+03 4.724235e+03
## factor(engine_type)Propjet:max_speed 3.465151e+03 7.929835e+03
## factor(engine_type)Piston:cruise_speed 3.633557e+03 1.300175e+04
## factor(engine_type)Propjet:cruise_speed -3.255795e+03 2.950109e+03
## factor(engine_type)Piston:wing_span 3.111048e+03 8.839655e+03
## factor(engine_type)Propjet:wing_span 3.532246e+03 8.402934e+03
## engine_power:max_speed -1.843571e+00 -6.958844e-01
## engine_power:cruise_speed -3.968461e+00 -2.457532e+00
## engine_power:wing_span 2.287495e-01 6.326622e-01
## max_speed:wing_span 1.833294e+00 1.944134e+01
## cruise_speed:stall_speed 2.422499e+01 7.418080e+01Our model identifies key drivers of aircraft price, with engine type, engine power, cruise speed, stall speed, and wing span playing significant roles. Engine power increases price, while stall speed and wing span tend to decrease it. Notably, Propjet aircraft benefit more from higher speeds and larger wingspans. The most significant predictors are cruise speed, engine power, and cruise speed squared. We are 95% confident that for every one-unit increase in cruise speed, price decreases by 1,660 to 13,300, while engine power increases price by 1,280 to 2,120 per unit. Additionally, the quadratic cruise speed term shows a nonlinear price effect, meaning price increases at higher cruise speeds but at a diminishing rate.
bestModel = lm(price ~ factor(engine_type) + engine_power + I(cruise_speed^2) +
max_speed + cruise_speed + stall_speed + wing_span +
factor(engine_type) * engine_power + factor(engine_type) * max_speed +
factor(engine_type) * cruise_speed + factor(engine_type) * wing_span +
engine_power * max_speed + engine_power * cruise_speed + engine_power * wing_span +
max_speed * wing_span + cruise_speed * stall_speed, data = AircraftData)
newdata = data.frame(engine_type = "Piston",engine_power=300, max_speed =240, cruise_speed =200, stall_speed=70, wing_span=400)
predict(bestModel,newdata,interval="predict",level = .95)## fit lwr upr
## 1 2324571 1765506 2883635The model predicts a fitted price of 2,324,571 USD for a piston-engine aircraft with the given specifications: engine_power = 300, max_speed = 240, cruise_speed = 200, stall_speed = 70, and wing_span = 400. The 95% prediction interval ranges from 1,765,506 USD to 2,883,635 USD.
The linear regression model assumes that there is a straight-line (linear) relationship between the predictors and the aircraft price. For identifying the non linearity we are using the residual plot.
#Residual Plot
bestModel = lm(price ~ factor(engine_type) + engine_power + I(cruise_speed^2) +
max_speed + cruise_speed + stall_speed + wing_span +
factor(engine_type) * engine_power + factor(engine_type) * max_speed +
factor(engine_type) * cruise_speed + factor(engine_type) * wing_span +
engine_power * max_speed + engine_power * cruise_speed + engine_power * wing_span +
max_speed * wing_span + cruise_speed * stall_speed, data = AircraftData)
summary(bestModel)$adj.r.squared## [1] 0.9247535sigma(bestModel)## [1] 279449.2ggplot(bestModel, aes(x=.fitted, y=.resid)) +
geom_point() + geom_smooth(color = "lightblue") +
geom_hline(yintercept = 0) +
labs(title = "Residual vs fitted values",
x = "Fitted",
y = "Residual",
color = "Method") +
theme_minimal()## `geom_smooth()` using method = 'loess' and formula = 'y ~ x'
There appears to be no pattern of the residuals at all. The 𝑅2𝑎𝑑𝑗 of the bestModel is 0.9247535 indicates the variation in Price that can be explained by this model is 92.48% with RMSE = 279449.2. We can conclude that the above model named bestModel, is the best fit model to predict Price among the models we considered.
We will now extract the residuals and plot this data to analyze the histogram, qq plot, and residual vs fitted plot.
residuals = resid(bestModel)
hist(residuals, main="Histogram of Residuals - Best Model", xlab="Residuals", col="lightblue", border="black", breaks = 30)
The histogram appears to be somewhat centered around zero, which is good. However, the shape does not perfectly follow a bell curve (normal distribution). The bars are somewhat skewed—there seems to be a longer right tail, suggesting potential positive skewness. Next, we will confirm whether the residuals deviate from normality, using the Q-Q Plot.
# Q-Q plot of residuals
qqnorm(residuals, main="Q-Q Plot of Residuals")
qqline(residuals, col="lightblue")
The points deviate significantly at both ends (left and right tails), indicating non-normality in the residuals. The right tail (higher values) has several points far above the line, suggesting the presence of high-value outliers (potentially high-priced aircraft affecting the model). The left tail (lower values) slightly deviates as well. Most points near the center follow the red line well, which means most residuals are approximately normal. Since the p-value is very small, your residuals do not follow a normal distribution. This violates the normality assumption in linear regression, which can impact hypothesis testing and confidence intervals.
# Plot Residuals vs Fitted Values
plot(fitted(bestModel), residuals, main="Residuals vs Fitted", xlab="Fitted Values", ylab="Residuals")
abline(h=0, col="red")
The Residuals vs. Fitted plot shows signs of heteroscedasticity, as residuals exhibit increasing variance for larger fitted values, indicating that the model struggles to predict high aircraft prices accurately. Additionally, a slight curvature suggests potential non-linearity, meaning the model might be missing key interaction or polynomial effects. There are also outliers that could be influential observations, impacting the model’s stability. These issues violate regression assumptions, potentially leading to unreliable standard errors and biased predictions.
Overall, this Residuals vs. Fitted plot confirms that your model suffers from heteroscedasticity and possible outliers. Applying log transformation, WLS regression, or robust regression techniques should improve the model’s reliability
In linear regression, the error terms (𝜖₁, 𝜖₂, 𝜖₃, etc.) should not be related to each other. This means the value of one error doesn’t give any clue about the next one. When this assumption is violated (e.g., errors are correlated), it often happens in time-series data, where observations are taken over time. As we do not have any time series data or spatial variable, we considered to plot the residuals vs engine_type category.
residuals <- residuals(bestModel)
residuals_data <- data.frame(Residuals = residuals, EngineType = AircraftData$engine_type)
boxplot(Residuals ~ EngineType, data = residuals_data,
main = "Residuals vs Engine Type",
xlab = "Engine Type",
ylab = "Residuals",
col = "lightblue")
The box plot above indicates that the residuals for different engine types are centered around zero and do not display any noticeable patterns. Therefore, the independence assumption is satisfied.
bestModel = lm(price ~ factor(engine_type) + engine_power + I(cruise_speed^2) +
max_speed + cruise_speed + stall_speed + wing_span +
factor(engine_type) * engine_power + factor(engine_type) * max_speed +
factor(engine_type) * cruise_speed + factor(engine_type) * wing_span +
engine_power * max_speed + engine_power * cruise_speed + engine_power * wing_span +
max_speed * wing_span + cruise_speed * stall_speed, data = AircraftData)
#residuals plot
plot(bestModel, which=1) 
#scale-location plot
plot(bestModel, which=3) 
#The Breusch-Pagan Test
bptest(bestModel)##
## studentized Breusch-Pagan test
##
## data: bestModel
## BP = 67.112, df = 21, p-value = 1.013e-06H0: heteroscedasticity is not present (homoscedasticity)
H1: heteroscedasticity is present
Here we reject the null hypothesis (p-value < 0.05), so we conclude we do have heteroscedasticity. Maybe we can try a higher order model to get rid of this issue.
To resolve we can apply log transformation
AircraftData$log_price = log(AircraftData$price)
bestModel_log <- lm(log_price ~ factor(engine_type) + engine_power + I(cruise_speed^2) +
max_speed + cruise_speed + stall_speed + wing_span +
factor(engine_type) * engine_power + factor(engine_type) * max_speed +
factor(engine_type) * cruise_speed + factor(engine_type) * wing_span +
engine_power * max_speed + engine_power * cruise_speed + engine_power * wing_span +
max_speed * wing_span + cruise_speed * stall_speed, data = AircraftData)
bptest(bestModel_log)##
## studentized Breusch-Pagan test
##
## data: bestModel_log
## BP = 124.51, df = 21, p-value < 2.2e-16plot(bestModel_log, which=3)
#sqrt transformation
AircraftData$sqrt_price = sqrt(AircraftData$price)
bestModel_sqrt <- lm(sqrt_price ~ factor(engine_type) + engine_power + I(cruise_speed^2) +
max_speed + cruise_speed + stall_speed + wing_span +
factor(engine_type) * engine_power + factor(engine_type) * max_speed +
factor(engine_type) * cruise_speed + factor(engine_type) * wing_span +
engine_power * max_speed + engine_power * cruise_speed + engine_power * wing_span +
max_speed * wing_span + cruise_speed * stall_speed, data = AircraftData)
bptest(bestModel_sqrt)##
## studentized Breusch-Pagan test
##
## data: bestModel_sqrt
## BP = 93.847, df = 21, p-value = 3.473e-11plot(bestModel_sqrt, which=3)
p-value < 2.2e-16 is less than 0.05, so we reject H0. Therefor there exists hetroscedasticity. Now we apply Box-Cox transformations.
bc=boxcox(bestModel,lambda=seq(-5,5))
bestlambda=bc$x[which(bc$y==max(bc$y))]
(bestlambda)## [1] 0.8585859#The Breusch-Pagan Test
bptest(bestModel)##
## studentized Breusch-Pagan test
##
## data: bestModel
## BP = 67.112, df = 21, p-value = 1.013e-06bcmodel=lm((((price^0.8585859)-1)/0.8585859) ~ factor(engine_type) + engine_power + I(cruise_speed^2) +
max_speed + cruise_speed + stall_speed + wing_span +
factor(engine_type) * engine_power + factor(engine_type) * max_speed +
factor(engine_type) * cruise_speed + factor(engine_type) * wing_span +
engine_power * max_speed + engine_power * cruise_speed + engine_power * wing_span +
max_speed * wing_span + cruise_speed * stall_speed, data = AircraftData)
#summary(bcmodel)
bptest(bcmodel)##
## studentized Breusch-Pagan test
##
## data: bcmodel
## BP = 71.935, df = 21, p-value = 1.714e-07The Box-Cox transformation (using λ = 0.8586) helped to adjust the dependent variable, but it did not completely solve the heteroscedasticity problem in the model.
bestModel = lm(price ~ factor(engine_type) + engine_power + I(cruise_speed^2) +
max_speed + cruise_speed + stall_speed + wing_span +
factor(engine_type) * engine_power + factor(engine_type) * max_speed +
factor(engine_type) * cruise_speed + factor(engine_type) * wing_span +
engine_power * max_speed + engine_power * cruise_speed + engine_power * wing_span +
max_speed * wing_span + cruise_speed * stall_speed, data = AircraftData)
plot(bestModel,which=5)
AircraftData[cooks.distance(bestModel)>0.5,]## model_name engine_type engine_power max_speed cruise_speed
## 47 100 Darter (S.L. Industries) Propjet 1050 158 126
## stall_speed fuel_tank all_eng_roc out_eng_roc takeoff_distance
## 47 52 216 650 1706 1525
## landing_distance empty_weight length wing_span range price log_price
## 47 12500 5829 402 438 600 8e+05 13.59237
## sqrt_price
## 47 894.4272#which =4 only prints the cook distance plot.
plot(bestModel,pch=18,col="lightblue",which=c(4)) 
#cook distance
AircraftData[cooks.distance(bestModel)>0.5,]## model_name engine_type engine_power max_speed cruise_speed
## 47 100 Darter (S.L. Industries) Propjet 1050 158 126
## stall_speed fuel_tank all_eng_roc out_eng_roc takeoff_distance
## 47 52 216 650 1706 1525
## landing_distance empty_weight length wing_span range price log_price
## 47 12500 5829 402 438 600 8e+05 13.59237
## sqrt_price
## 47 894.4272#leverages points plots
plot(bestModel,pch=18,col="lightblue",which=c(4)) #which =4 only prints the cook distance plot.From here you can see that data point 47 is an influential case and we need to work for that point so find cook distance. Observation 47 (0.65) well above this threshold, indicating they are highly influential points.
lev=hatvalues(bestModel)
p = length(coef(bestModel))
n = nrow(AircraftData)
outlier2p = lev[lev>(2*p/n)]
outlier3p = lev[lev>(3*p/n)]
print("h_I>2p/n, outliers are")## [1] "h_I>2p/n, outliers are"print(outlier2p)## 14 45 47 48 49 50 153
## 0.11211970 0.19426975 0.43469053 0.13561182 0.09810647 0.09139571 0.10206065
## 154 155 156 175 176 177 179
## 0.37351354 0.30238020 0.20935904 0.79251544 0.26615958 0.33245314 0.32092689
## 181 184 185 186 187 188 189
## 0.17178242 0.14374943 0.14654533 0.14654533 0.26874702 0.16242552 0.08810582
## 190 191 192 194 305 362 377
## 0.10251860 0.11437497 0.11659846 0.12811915 0.11543268 0.14168106 0.11898507
## 378 379 388 389 390 400 401
## 0.12871465 0.12417699 0.19259490 0.18877506 0.23889980 0.11180028 0.52735558
## 409 412 417 418 419 423 424
## 0.20863546 0.10646193 0.11603622 0.13688508 0.10649390 0.78416687 0.21189792
## 425 426 427 428 429 430 450
## 0.19516285 0.11945897 0.11672524 0.13734294 0.21329228 0.54310889 0.12264173
## 468 469 480 482 511 512 513
## 0.28983059 0.09922146 0.10850892 0.88392524 0.42966304 0.10650121 0.12815809print(outlier3p)## 45 47 48 154 155 156 175 176
## 0.1942698 0.4346905 0.1356118 0.3735135 0.3023802 0.2093590 0.7925154 0.2661596
## 177 179 181 184 185 186 187 188
## 0.3324531 0.3209269 0.1717824 0.1437494 0.1465453 0.1465453 0.2687470 0.1624255
## 362 388 389 390 401 409 418 423
## 0.1416811 0.1925949 0.1887751 0.2388998 0.5273556 0.2086355 0.1368851 0.7841669
## 424 425 428 429 430 468 482 511
## 0.2118979 0.1951629 0.1373429 0.2132923 0.5431089 0.2898306 0.8839252 0.4296630Here 47 leverage point is 0.4346 and cook distance 0.65 so this is high influential points and remove and check the result.
#ouliers removed
AircraftData_NoOutliers = read.csv("/Users/anithajoseph/Documents/aircraft_price_no_outliers.csv")
sum(is.na(AircraftData_NoOutliers))## [1] 10Aircraft_NoOutliers = na.omit(AircraftData_NoOutliers)
bestModel_NoOutliers = lm(price ~ factor(engine_type) + engine_power + I(cruise_speed^2) +
max_speed + cruise_speed + stall_speed + wing_span +
factor(engine_type) * engine_power + factor(engine_type) * max_speed +
factor(engine_type) * cruise_speed + factor(engine_type) * wing_span +
engine_power * max_speed + engine_power * cruise_speed + engine_power * wing_span +
max_speed * wing_span + cruise_speed * stall_speed, data = Aircraft_NoOutliers)
summary(bestModel_NoOutliers)$adj.r.square## [1] 0.9273244sigma(bestModel_NoOutliers)## [1] 274264.5summary(bestModel)$adj.r.square## [1] 0.9247535sigma(bestModel)## [1] 279449.2Our best model predicts aircraft price based on engine type, performance metrics such as cruise speed, engine power, stall speed, max speed, and design factors like wing span. On analysis of the best model, we can see the below findings:
Engine Type:
Jet type is the reference category
Piston price is less than Jet by approximately 4.39 million USD
Propjet price is less than Jet by approximately 2.76 million USD
Cruise Speed impact:
Linear term: negative effect on price
Quadratic term: positive effect with exponential price increase at higher speeds
Negative Effects: for stall speed and wing span
Interaction terms:
For Propjets, Higher max speed benefits, but higher cruise speed reduces price.
Engine Power + Max Speed / Cruise Speed: Slight negative price impact
Max Speed + Wing Span: Contributes positively to price
Jet Engines: Predicted price is 5,823,241.56 USD when all other variables are zero. Jets do not experience price reductions like piston engines or propjets. Negative effects come from stall speed and wing span, with mixed interactions across other factors.
Piston Engines: Predicted price is about 4.39 million USD lower than jets. Engine power decreases price by 981 units per unit increase, while max speed and cruise speed positively impact price. Greater wing span also increases price, but stall speed and wing span have overall negative effects.
PropJets Engines: Predicted price is approximately 2.76 million USD lower than jets. Engine power reduces price by 2,284.54 units per unit increase, but max speed and wing span increase price. Higher cruise speed slightly decreases price, while stall speed and wing span have negative effects.
Since the best model has not meet with homoscedasticity or equal variance assumption. It could be resolved by doing some of the following methods
Given that the predictors engine_power and cruise_speed are statistically and theoretically significant for our dataset, we chose to retain them despite their VIF being greater than 5. While these terms hold theoretical importance, removing them might yield alternative models capable of predicting prices effectively. Further analysis is necessary to evaluate their relevance, and we suggest examining whether the newly developed model could address the heteroscedasticity problem.